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For some reason, piping this simple trumpet (which selects 'span' but doesn't do anything with them) to stdout works fine, but piping it to a file writestream, it doesn't reach the end...
Raw
a-trumpet-problem.js
var trumpet = require('../');
var through = require('through2');
var fs = require('fs');
trumpetBigassFile(process.stdout, "******* success with stdout!");
trumpetBigassFile(fs.createWriteStream('temp.html'), "******* success with file!");
function trumpetBigassFile(writeStream, message) {
var tr = trumpet();
tr.selectAll('span', function(elem){
// do nothing
});
tr.pipe(writeStream);
tr.on('end', function(){
console.log(message);
});
fs.createReadStream(__dirname + '/bigass-file.html').pipe(tr);
}
Raw
bigass-file.html
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="content-type" content="text/html; charset=utf-8" />
<title>
01-3-arithmetic-sequences-and-series.html
</title>
<style type="text/css" media="screen">
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color : #000000;
text-indent : 0px;
margin : 15px 0px;
}
p.Development---Problem---Single-Digit {
font-family : "Garamond Regular", sans-serif;
font-weight : normal;
font-style : normal;
font-size : 1em;
text-decoration : none;
font-variant : normal;
line-height : 1.25;
text-align : left;
color : #000000;
text-indent : -22px;
margin : 15px 36px 15px 54px;
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</head>
<body>
<div id="x01-3-arithmetic-sequences-and-series.html" xml:lang="en-US" lang="en-US">
<div class="story">
<p class="Lesson---Title">
lesson 3: Arithmetic Sequences and Series
</p>
<p class="Lesson---Section-title">
Introduction
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">1</span> Fill in the blanks to continue the pattern you see in this list: <img class="image" width="171" height="16" src="01-3-arithmetic-sequences-and-series-web-images/Eqn001.eps" alt="Eqn001.eps" />
</p>
<p class="Lesson---Narrative-text">
In mathematics, a <span class="char-style-override-2">sequence</span> is a list of numbers, usually with an identifiable<br />
pattern or a connection to real data — for example, the list of expected<br />
temperatures for the next ten days, or a sequence listing the profits that a<br />
business earned year by year.
</p>
<p class="Lesson---Narrative-text">
The numbers in the sequence are called <span class="char-style-override-2">terms</span> — for example, in the sequence above, 2 is the first term. Sometimes this pattern is very simple: 2,4,6,8…. Other times it might be harder to spot, like the problem above. Or, it might be an<br />
unusual type of pattern — 0,1,0,1,1,0,1,1,1,0,1,1,1,1….
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">2</span> Come up with the next two numbers in the sequence <img class="image" width="56" height="14" src="01-3-arithmetic-sequences-and-series-web-images/Eqn002.eps" alt="Eqn002.eps" />, and a reason for why they should be the two numbers.
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">3</span> How about the next two numbers in the sequence <img class="image" width="93" height="14" src="01-3-arithmetic-sequences-and-series-web-images/Eqn0031.eps" alt="Eqn0031.eps" />?
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">4</span> Make up a sequence that you think will challenge your neighbor — then see if they can find the pattern.
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">5</span> A sequence starts with the numbers 5, 15, 35.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. Describe a pattern that this sequence might be following, and give the next three terms according to the pattern.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. Now describe a different pattern that the sequence might be following, and give the next three terms according to the new pattern.
</p>
<p class="Lesson---Narrative-text">
The patterns you’ve seen so far have often been related to the differences<br />
between terms, such as the patterns in problems 1 and 2. When the pattern in a sequence is simply that the difference is always the same, the sequence is called an <span class="char-style-override-2">arithmetic sequence</span>. For example, 7,9,11,13… is one of these.
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">6</span> Some of the sequences below are arithmetic, and some are not. For each sequence, only a few terms are given. Identify<br />
whether or not the sequence appears to be arithmetic. Also for each sequence find a pattern and fill in the missing terms.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. 3, 6, 12, 24, ___, ___, …
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. 4, 11, 18, ___, ___, ….
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">c</span>. 9, ___, 17, ___, 25, ….
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">d</span>. 12, 13, 15, 16, 18, ___, ___,….
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">e</span>. 47, 43, 39, ___, 31, ___, ___, ….
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">f</span>. 13, 16, 22, 24, 28, 36, ___, ___, ___....
</p>
<p class="Lesson---Section-title"></p>
<p class="Lesson---Section-title">
Development
</p>
<p class="Lesson---Narrative-text">
In a sequence, it’s not only the numbers that matter—the order of the numbers is important too. For instance, sequence “A” — 20,10,40,30,60,50… — follows quite a different pattern from sequence “B” — 10,20,30,40,50,60…. — even though they contain all the same numbers.
</p>
<p class="Lesson---Narrative-text">
In the first sequence, 30 is the 4th term, while in the second sequence, 30 is<br />
the 3rd term. To make this type of thing easier to write, we’ll introduce some notation.
</p>
<p class="Lesson---Narrative-text">
In the first sequence, we write “<img class="image" width="45" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn004.eps" alt="Eqn004.eps" />” (pronounced “A sub four” or simply “A four”) to mean that the sequence’s 4th term is 30. Then the first sequence is called <img class="image" width="26" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn005.eps" alt="Eqn005.eps" /> (because it is a set that consists of the terms <img class="image" width="17" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn006.eps" alt="Eqn006.eps" />, <img class="image" width="18" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn007.eps" alt="Eqn007.eps" />, <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn008.eps" alt="Eqn008.eps" />, etc) and the second sequence is called <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn009.eps" alt="Eqn009.eps" />.
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">7</span> Fill in the blanks according to the sequences above.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. <img class="image" width="17" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn010.eps" alt="Eqn010.eps" />=_____.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. <img class="image" width="15" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn011.eps" alt="Eqn011.eps" />=_____.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">c</span>. <img class="image" width="29" height="21" src="01-3-arithmetic-sequences-and-series-web-images/Eqn012.eps" alt="Eqn012.eps" />= 30.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">d</span>. <img class="image" width="33" height="21" src="01-3-arithmetic-sequences-and-series-web-images/Eqn013.eps" alt="Eqn013.eps" />= 70.
</p>
<p class="Lesson---Narrative-text para-style-override-1">
One last piece of notation you’ll need to know is that, in an arithmetic sequence, the constant difference between terms is often called “<span class="char-style-override-3">d</span>”. In the sequence below, <img class="image" width="30" height="13" src="01-3-arithmetic-sequences-and-series-web-images/Eqn014.eps" alt="Eqn014.eps" />.<br />
7, 9, 11, 13, . . .
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">8</span> In an arithmetic sequence called <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn015.eps" alt="Eqn015.eps" />, if <img class="image" width="39" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn016.eps" alt="Eqn016.eps" />, and <img class="image" width="29" height="13" src="01-3-arithmetic-sequences-and-series-web-images/Eqn017.eps" alt="Eqn017.eps" />, then what does <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn018.eps" alt="Eqn018.eps" /> equal? What does <img class="image" width="22" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn019.eps" alt="Eqn019.eps" /> equal?
</p>
<p class="Development---Problem---Single-Digit">
<span class="Problem-Number">9</span> In an arithmetic sequence, <img class="image" width="35" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn020.eps" alt="Eqn020.eps" /> and <img class="image" width="39" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn021.eps" alt="Eqn021.eps" />. Find the first three terms of the sequence.
</p>
<p class="Development---Problems">
<span class="Problem-Number">10</span> An arithmetic sequence starts with <img class="image" width="35" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn022.eps" alt="Eqn022.eps" />, <img class="image" width="35" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn023.eps" alt="Eqn023.eps" />, … Which term will equal 99?
</p>
<p class="Development---Problems">
<span class="Problem-Number">11</span> What might the next two terms be in the sequence 4, 6, 9…? Find several different possible patterns.
</p>
<p class="Development---Problems">
<span class="Problem-Number">12</span> Make up five sequences all beginning with 1, 2, but with a different third term. Look at what others in your class came up with, as seeing different patterns can give you ideas for making up some really interesting sequences.
</p>
<p class="Development---Problems">
<span class="Problem-Number">13</span> Consider the sequence 5, 10, 6, 12, 8, 16, 12, 24, ...
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. Find the repeating pattern, and write the next 3 terms.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. It looks like all of the numbers after the “5” are even —<br />
will the sequence ever produce another odd number?<br />
Why or why not?
</p>
<p class="Development---Problems">
<span class="Problem-Number">14</span> Consider the three sequences below. Sequences <img class="image" width="25" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn024.eps" alt="Eqn024.eps" /> and <img class="image" width="25" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn025.eps" alt="Eqn025.eps" /> below have some missing information, but you can assume that they are arithmetic.
</p>
<table id="table-1" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
1
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
2
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
3
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn026.eps" alt="Eqn026.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
12
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
19
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
26
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
33
</p>
</td>
</tr>
</tbody>
</table>
<table id="table-2" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
2
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
6
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
8
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
10
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="17" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn027.eps" alt="Eqn027.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
?
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
41
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
?
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
48
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
?
</p>
</td>
</tr>
</tbody>
</table>
<p class="Development---Problem-Flow"></p>
<table id="table-3" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
9
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
11
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
13
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
15
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
17
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="16" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn028.eps" alt="Eqn028.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
?
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
6
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
?
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
34
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
?
</p>
</td>
</tr>
</tbody>
</table>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. Which sequence, <img class="image" width="25" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn029.eps" alt="Eqn029.eps" /> or <img class="image" width="25" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn030.eps" alt="Eqn030.eps" />, has the same value of <span class="char-style-override-3">d</span> as sequence <img class="image" width="26" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn031.eps" alt="Eqn031.eps" />?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. Plot the information from the three sequences on a graph (put <span class="char-style-override-3">n</span> on the horizontal axis). Put all three on the same graph. How does what you see fit with your answer to<br />
part a?
</p>
<p class="Development---Problems">
<span class="Problem-Number">15</span> In your thermometer, the mercury reaches a height of 46 mm when the temperature is 70 degrees Fahrenheit, and a height of 48 mm when the temperature is 73 degrees.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. Think about these data as an arithmetic sequence where n is the temperature, and <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn032.eps" alt="Eqn032.eps" /> is the height at that temperature. Note that “…” is used here to indicate that this table is not showing you values of <span class="char-style-override-3">n</span> and <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn033.eps" alt="Eqn033.eps" /> where <span class="char-style-override-3">n</span> is less than 70 or greater than 75, but such values still exist.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">F</span>ill in the table below:
</p>
<table id="table-4" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
70
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
71
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
72
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
73
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
74
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
75
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn034.eps" alt="Eqn034.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
46
</p>
</td>
<td class="Basic-cell"></td>
<td class="Basic-cell"></td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
48
</p>
</td>
<td class="Basic-cell"></td>
<td class="Basic-cell"></td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. What is the value of <span class="char-style-override-3">d</span> for this sequence?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">c</span>. What would the height be if the temperature were at 106 degrees Fahrenheit?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">d</span>. What would the height be if the temperature were at 65 degrees Fahrenheit?
</p>
<p class="Development---Problems">
<span class="Problem-Number">16</span> In an arithmetic sequence, the <span class="char-style-override-4">constant</span> difference is sometimes referred to as the rate of change of the sequence. Any idea why it might be so called?
</p>
<p class="Development---Problems">
<span class="Problem-Number">17</span> If a sequence has <img class="image" width="41" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn035.eps" alt="Eqn035.eps" /> but it has a rate of change of -6, find <img class="image" width="15" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn036.eps" alt="Eqn036.eps" />, <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn037.eps" alt="Eqn037.eps" />, and <img class="image" width="19" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn038.eps" alt="Eqn038.eps" />.
</p>
<p class="Development---Problems">
<span class="Problem-Number">18</span> How would you find the value of <span class="char-style-override-3">d</span> for the arithmetic<br />
sequence below? (Remember that “…” is used to indicate that there are other value(s) of <span class="char-style-override-3">n</span> and <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn039.eps" alt="Eqn039.eps" /> that are not listed in this table but certainly exist!) Also write an equation for <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn040.eps" alt="Eqn040.eps" /> in terms of <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn041.eps" alt="Eqn041.eps" />, <span class="char-style-override-3">a</span>, <span class="char-style-override-3">b</span> and <span class="char-style-override-3">d</span>.
</p>
<table id="table-5" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">a</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">b</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn042.eps" alt="Eqn042.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="char-style-override-5"><img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn043.eps" alt="Eqn043.eps" /></span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="char-style-override-5"><img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn044.eps" alt="Eqn044.eps" /></span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Lesson---Section-title">
Practice
</p>
<p class="Development---Problems">
<span class="Problem-Number">19</span> Another thermometer has a different relationship between height and temperature. For this thermometer, the equation <img class="image" width="84" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn045.eps" alt="Eqn045.eps" /> expresses the height <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn046.eps" alt="Eqn046.eps" /> (in mm) of the mercury in terms of the temperature <span class="char-style-override-3">n</span>, in degrees Fahrenheit.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. What’s the height when the temperature is 50 degrees?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. If you think of <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn047.eps" alt="Eqn047.eps" /> as the nth term of a sequence, what’s the first term?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">c</span>. What’s the value of <span class="char-style-override-3">d</span> for the sequence <img class="image" width="27" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn048.eps" alt="Eqn048.eps" />?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">d</span>. What would be the temperature if the height of the mercury were 20.7 mm ?
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">e</span>. What would be the temperature if the height of the mercury were 3 mm?
</p>
<p class="Development---Problems">
<span class="Problem-Number">20</span> An arithmetic sequence <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn049.eps" alt="Eqn049.eps" /> has <img class="image" width="37" height="13" src="01-3-arithmetic-sequences-and-series-web-images/Eqn050.eps" alt="Eqn050.eps" />, and <img class="image" width="18" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn051.eps" alt="Eqn051.eps" /> = 48.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>. Find <img class="image" width="15" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn052.eps" alt="Eqn052.eps" />.
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>. For what value of <span class="char-style-override-3">n</span> would <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn053.eps" alt="Eqn053.eps" /> = 130.5?<br />
</p>
<p class="Development---Problems">
<span class="Problem-Number">21</span> Each part below gives information about a sequence <img class="image" width="26" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn054.eps" alt="Eqn054.eps" />.<br />
In each part, find the value of the unknown in the table for <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn055.eps" alt="Eqn055.eps" />so that the two sequences have the same rate of change.
</p>
<p class="Development---Problem-Flow">
(Remember that “…” is used to indicate that there are other value(s) of <span class="char-style-override-3">n</span> , <img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn056.eps" alt="Eqn056.eps" />, and <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn057.eps" alt="Eqn057.eps" /> that are not listed in a table but certainly exist. Thus in the first table of part a., <span class="char-style-override-3">n</span> can be less than 6, can be 7 or 8, and can be greater than 9, even though those values of <span class="char-style-override-3">n</span> aren’t listed explicitly in the table.)
</p>
<p class="Development---Problem-Part">
<span class="char-style-override-1">a</span>.
</p>
<table id="table-6" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
6
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
9
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn058.eps" alt="Eqn058.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
40
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
82
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<table id="table-7" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
5
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn059.eps" alt="Eqn059.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">a</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
48
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Development---Problem-Part">
<span class="char-style-override-1">b</span>.
</p>
<table id="table-8" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
16
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
22
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn060.eps" alt="Eqn060.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
50
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
23
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<table id="table-9" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
1
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
19
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn061.eps" alt="Eqn061.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">b</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
40
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Development---Problem-Part">
<span class="char-style-override-1">c</span>.
</p>
<table id="table-10" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
9
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
31
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn062.eps" alt="Eqn062.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
40.5
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
71.3
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<table id="table-11" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
14
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn063.eps" alt="Eqn063.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">c</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Development---Problem-Part">
<span class="char-style-override-1">d</span>.
</p>
<table id="table-12" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
10
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
12
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn064.eps" alt="Eqn064.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
12
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
20
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<table id="table-13" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
30
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">q</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn065.eps" alt="Eqn065.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
9
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
29
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
</div>
<div class="Basic-Text-Frame">
<p class="Lesson---Page-Number">
41
</p>
</div>
<div class="Basic-Text-Frame">
<p class="Lesson---Page-Number para-style-override-2"></p>
</div>
<div class="Basic-Text-Frame">
<p class="Lesson---Header para-style-override-2">
Book 1: Reasoning and Proving I
</p>
</div>
<div class="Basic-Text-Frame">
<p class="Lesson---Header">
Lesson 3: ARITHMETIC SEQUENCES AND SERIES
</p>
</div>
<div class="Basic-Text-Frame">
<p class="Lesson---Page-Number"></p>
</div>
<div class="Basic-Text-Frame">
<p class="Basic-Paragraph">
<span class="char-style-override-3">(continued on next page)</span>
</p>
</div>
<div class="Basic-Text-Frame">
<p class="Problems---Problem">
<span class="Problem-Number">22</span> A sequence <img class="image" width="26" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn066.eps" alt="Eqn066.eps" /> increases by 7 every 5 terms. Starting at the 2009th term, how far would you have to go to increase by a total of 2009?
</p>
<p class="Problems---Problem">
<span class="Problem-Number">23</span> In an arithmetic sequence…
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. The first term is 10, and the difference is <span class="char-style-override-3">d</span>. In terms of <span class="char-style-override-3">d</span>, what is the 20th term? How would you get the <span class="char-style-override-3">n</span>th term?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. The first term is <img class="image" width="13" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn067.eps" alt="Eqn067.eps" />, and the difference is <span class="char-style-override-3">d</span>. What is the 20th term? How would you get the <span class="char-style-override-3">n</span>th term?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. Write an equation for the <span class="char-style-override-3">n</span>th term,<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn068.eps" alt="Eqn068.eps" /> in terms of <img class="image" width="13" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn069.eps" alt="Eqn069.eps" />, <span class="char-style-override-3">n</span> and <span class="char-style-override-3">d</span>.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">24</span> Suppose you have saved $35 and you also earn $15 for mowing a lawn.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. How many lawns will you have to mow in order to have $95 saved?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Suppose you wanted to save $380. How many lawns do you have to mow, and how did you get your answer?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. How about for $395?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">d</span>. Let n be the number of lawns you mow, and <img class="image" width="19" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn070.eps" alt="Eqn070.eps" /> be the amount of money you have saved after mowing those lawns. Find <img class="image" width="18" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn071.eps" alt="Eqn071.eps" /> <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn072.eps" alt="Eqn072.eps" />, and <img class="image" width="27" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn073.eps" alt="Eqn073.eps" />.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">e</span>. Write an equation for <img class="image" width="19" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn074.eps" alt="Eqn074.eps" /> in terms of <span class="char-style-override-3">n</span>.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">25</span> In an arithmetic sequence <img class="image" width="24" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn075.eps" alt="Eqn075.eps" />, you know the following values: <img class="image" width="41" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn076.eps" alt="Eqn076.eps" />, <img class="image" width="42" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn077.eps" alt="Eqn077.eps" />, <img class="image" width="42" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn078.eps" alt="Eqn078.eps" />.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. What’s the rate of change?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Does the equation “<img class="image" width="66" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn079.eps" alt="Eqn079.eps" />”<br />
hold true for the sequence? If yes, explain why it works, and why it will work for every value of <span class="char-style-override-3">n</span>. If no, explain why it fails, and fix it to make it work.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">26</span> The graph below is a data plot for an arithmetic sequence <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn080.eps" alt="Eqn080.eps" />.<br />
<br />
</p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. What is the rate of change of <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn081.eps" alt="Eqn081.eps" />?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Write an equation for the value of <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn082.eps" alt="Eqn082.eps" /> in terms of <span class="char-style-override-3">n</span>.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">27</span> Using the information from sequence <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn0531.eps" alt="Eqn053.eps" /> below, do the following:
</p>
<table id="table-14" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
1
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
2
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
3
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="17" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn084.eps" alt="Eqn084.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
4
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
10
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
16
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
22
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. Write an equation for the nth term of the arithmetic sequence <img class="image" width="24" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn085.eps" alt="Eqn085.eps" /> above, in terms of <span class="char-style-override-3">n</span> (<img class="image" width="41" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn086.eps" alt="Eqn086.eps" />). Be sure to test your equation by plugging in a few values of <span class="char-style-override-3">n</span> to make sure that it works.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Use your equation to determine what <img class="image" width="16" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn087.eps" alt="Eqn087.eps" /> will be when <img class="image" width="35" height="13" src="01-3-arithmetic-sequences-and-series-web-images/Eqn088.eps" alt="Eqn088.eps" />.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. Use your equation to determine what value of <span class="char-style-override-3">n</span> would make <img class="image" width="42" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn089.eps" alt="Eqn089.eps" />.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">28</span> Suppose for an arithmetic sequence <img class="image" width="23" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn090.eps" alt="Eqn090.eps" />, <img class="image" width="13" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn091.eps" alt="Eqn091.eps" /> = 378 and <span class="char-style-override-3">d</span> = .001.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. Without using a calculator, roughly estimate what <span class="char-style-override-3">n</span> would have to be so that <img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn092.eps" alt="Eqn092.eps" /> would be about 1 billion.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Now find the answer with a calculator and check that your estimate was a reasonable one.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">29</span> Write an equation for an arithmetic<br />
sequence whose rate of change is 8,<br />
and whose fourth term is 60.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">30</span> Find <span class="char-style-override-3">x</span> if <img class="image" width="69" height="16" src="01-3-arithmetic-sequences-and-series-web-images/Eqn093.eps" alt="Eqn093.eps" /> is an arithmetic sequence.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">31</span> Find <span class="char-style-override-3">x</span> if <img class="image" width="48" height="16" src="01-3-arithmetic-sequences-and-series-web-images/Eqn094.eps" alt="Eqn094.eps" /> is an arithmetic<br />
sequence.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">32</span> When you give <span class="char-style-override-6">♣</span>(a,b) two consecutive terms, <span class="char-style-override-3">a</span> and <span class="char-style-override-3">b</span>, of an arithmetic sequence, it gives you the next number in the<br />
sequence. Write an equation for <span class="char-style-override-6">♣</span>(a,b).
</p>
<p class="Problems---Problem">
<span class="Problem-Number">33</span> Create three arithmetic sequences where <img class="image" width="40" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn095.eps" alt="Eqn095.eps" /> and at least one is a decreasing sequence. Looking at your findings, is there any relationship between <img class="image" width="13" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn096.eps" alt="Eqn096.eps" /> and the constant difference <span class="char-style-override-3">d</span>? Explain.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">34</span> Do the sequences with the following <span class="char-style-override-3">n</span>th terms have any terms in common? <img class="image" width="66" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn097.eps" alt="Eqn097.eps" /> and <img class="image" width="72" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn098.eps" alt="Eqn098.eps" />.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">35</span> For a sequence <img class="image" width="24" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn099.eps" alt="Eqn099.eps" />, <img class="image" width="19" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn100.eps" alt="Eqn100.eps" />= 100, and<br />
<img class="image" width="39" height="14" src="01-3-arithmetic-sequences-and-series-web-images/Eqn003.eps" alt="Eqn003.eps" />.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. What is <img class="image" width="15" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn101.eps" alt="Eqn101.eps" />?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. How can this problem help explain why a negative times a negative is a positive?
</p>
<p class="Problems---Problem">
<span class="Problem-Number">36</span> Find <span class="char-style-override-3">x</span> if the sequence is arithmetic: 7, <span class="char-style-override-3">x</span>, 15. Do the same for 3, <span class="char-style-override-3">x</span>, 147. What would be a plausible description of what <span class="char-style-override-3">x</span> actually represents in each case? Would this always work when <span class="char-style-override-3">x</span> is the 2nd of three numbers?
</p>
<p class="Lesson---Narrative-text">
You get a <span class="char-style-override-2">series</span> when you add up the terms of a sequence. For example if you add the first four terms of 3,8,13,18,23,28,…. you get the series<br />
3 + 8 + 13 + 18. In the following problems,<br />
you’ll develop clever ways to find the sum of a<br />
series without actually having to add every term<br />
individually — this way, if you have to add up a series that’s 200 terms long, you won’t have to<br />
type 200 numbers into your calculator.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">37</span> Think of ways you could add up the series 1 + 2 + 3 + ... + 59 + 60 without having to actually add sixty numbers.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">38</span> Test out your strategies on the following series, then actually add them up number by number to check your answers:
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. 1 + 2 + 3 + ... + 10
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. 1 + 2 + 3 + ... + 15
</p>
<p class="Problems---Problem">
<span class="Problem-Number">39</span> How is the series 4 + 8 + 12 + ... + 236 + 240 related to the problems above?<br />
Calculate the sum of this new series.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">40</span> How about 5 + 8 + 11 + ... + 326 + 329? Find one way of calculating the series based on your work above, and one way of calculating the series without using your work above. Describe your strategies (and make sure they both give you the same answer!).
</p>
<p class="Problems---Problem">
<span class="Problem-Number">41</span> Using any strategy, find a quick way to calculate:
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. 1 + 2 + 3 + 4 + 5 + ... + 299 + 300
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. 3 + 4 + 5 + 6 + ... + 89 + 90 + 91
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. 6 + 8 + 10 + ... + 140
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">d</span>. 2 + 9 + 16 + ... + 72
</p>
<p class="Problems---Problem">
<span class="Problem-Number">42</span> Don’t use a calculator for this problem.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. Find <span class="char-style-override-4">5</span> + 3 <span class="char-style-override-4">1</span>
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Find 7 ÷ <span class="char-style-override-4">2</span>
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. Factor <span class="char-style-override-3"><img class="image" width="62" height="17" src="01-3-arithmetic-sequences-and-series-web-images/Eqn24.eps" alt="Eqn24.eps" /></span>
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">d</span>. Factor <span class="char-style-override-3"><img class="image" width="34" height="17" src="01-3-arithmetic-sequences-and-series-web-images/Eqn25.eps" alt="Eqn25.eps" /></span>
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">e</span>. Recall that |<span class="char-style-override-3">x</span>|, read “absolute<br />
value of <span class="char-style-override-3">x</span>,” means to make <span class="char-style-override-3">x</span><br />
positive, so |3|=3 and |-3|=3.<br />
Find |1-|3-|1-5<span>|||</span>.
</p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem">
<span class="Problem-Number">43</span> Johnny said that when he added up the 100 terms of the arithmetic series 123 + 129 + 135 + ... + 717 and got about 13000. Mary added up the same series and got about 69000. Halle added them up and got about 42000. One of them is clearly closest to the right answer — figure out which one it must be in under 30 seconds.
</p>
<p class="Problems---Problem">
<span class="Problem-Number">44</span> Interpret the diagram below as a series (imagine the number of dots in each piece is the next “term”, and add a few more pieces if necessary to see the pattern). Describe, in words, the terms being added, and describe the answers that you get as you continue adding terms.
</p>
<p class="Lesson---Inline-Diagram">
<img class="image" width="192" height="192" src="01-3-arithmetic-sequences-and-series-web-images/image007.tif" alt="image007.tif" />
</p>
<p class="Problems---Problem">
<span class="Problem-Number">45</span> Imagine writing out the first twenty counting numbers: 1, 2, 3, ... 18, 19, 20. Separate the sequence into odds and evens. Find the sum of the odd terms (if you have done problem 44, you’ll know the answer right off). Now consider how the sum of the even numbers compare to the sum of the odd numbers; and then see if you can generate a formula for adding<br />
the numbers 1-20; 1-100; 1-2<span class="char-style-override-3">n</span>. Anything familiar about your findings?
</p>
<p class="Problems---Problem">
<span class="Problem-Number">46</span> You add up the first n terms of an<br />
arithmetic sequence: <span class="char-style-override-3">T</span><span class="char-style-override-5">1</span>+...+<span class="char-style-override-3">T</span><span class="char-style-override-7">n</span>.
</p>
<p class="Development---Problem-Flow">
a. Use one of your strategies to give an equation for the answer, in terms of the<br />
variables <span class="char-style-override-3">T</span><span class="char-style-override-5">1</span>, <span class="char-style-override-3">T</span><span class="char-style-override-7">n</span>, and <span class="char-style-override-3">n</span>.
</p>
<p class="Development---Problem-Flow">
b. If <span class="char-style-override-3">d</span> is the difference of terms in the sequence,<br />
rewrite your equation in terms of <span class="char-style-override-3">T</span><span class="char-style-override-5">1</span>, <span class="char-style-override-3">d</span>, and <span class="char-style-override-3">n</span><br />
(without using <span class="char-style-override-3">T</span><span class="char-style-override-7">n</span> ).
</p>
<p class="Problems---Problem">
<span class="Problem-Number">47</span> An arithmetic sequence has <span class="char-style-override-3">T</span><span class="char-style-override-5">1</span>= 7. You know that <span class="char-style-override-3">d</span> is any integer from 1 to 20. What’s the probability that <span class="char-style-override-3">T</span><span class="char-style-override-7">n</span> = 27 for some <span class="char-style-override-3">n</span>?
</p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem">
<span class="Problem-Number">48</span> Compare the following two arithmetic sequences, {<span class="char-style-override-3">A</span><span class="char-style-override-7">n</span>} and <span><img class="image" width="22" height="19" src="01-3-arithmetic-sequences-and-series-web-images/Eqn26.eps" alt="Eqn26.eps" /></span>:
</p>
<table id="table-15" class="No-Table-Style table-style-override-1">
<tbody>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<span class="Variable">n</span>
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
1
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
2
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
3
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="18" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn1191.eps" alt="Eqn1191.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
40
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
44
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
48
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
<tr>
<td class="Basic-cell">
<p class="Table---Cell-Text">
<img class="image" width="14" height="20" src="01-3-arithmetic-sequences-and-series-web-images/Eqn1201.eps" alt="Eqn1201.eps" />
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
2
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
8
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
14
</p>
</td>
<td class="Basic-cell">
<p class="Table---Cell-Text">
…
</p>
</td>
</tr>
</tbody>
</table>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. Will sequence {<span class="char-style-override-3">B</span><span class="char-style-override-7">n</span>} ever catch up with sequence {<span class="char-style-override-3">A</span><span class="char-style-override-7">n</span>}? How do you know?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Write an equation for <span class="char-style-override-3">A</span><span class="char-style-override-7">n</span>, and an equation for <span class="char-style-override-3">B</span><span class="char-style-override-7">n</span>. Check that your equations work.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. Graph both of your equations on a calculator (“<span class="char-style-override-3">x</span>” can stand for “<span class="char-style-override-3">n</span>”) and use your graph to determine whether or not {<span class="char-style-override-3">B</span><span class="char-style-override-7">n</span>} ever catches up with {<span class="char-style-override-3">A</span><span class="char-style-override-7">n</span>}, and when.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">d</span>. Can you confirm that your answer in part c) is right, just by using the equations and NOT the graph?
</p>
<p class="Problems---Problem"></p>
<p class="Problems---Problem">
<span class="Problem-Number">49</span> Consider the sequence <img class="image" width="91" height="32" src="01-3-arithmetic-sequences-and-series-web-images/Eqn1271.eps" alt="Eqn1271.eps" />.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">a</span>. Find a pattern and the next two terms.
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">b</span>. Does the pattern consist of<br />
adding <span><img class="image" width="11" height="32" src="01-3-arithmetic-sequences-and-series-web-images/Eqn1281.eps" alt="Eqn1281.eps" /></span> to each term?
</p>
<p class="Problems---Problem-Part">
<span class="char-style-override-1">c</span>. Look at the ratios of each<br />
consecutive pair of terms —<br />
what do you find?
</p>
</div>
<div class="image">
<img class="media-1" width="217" height="216" src="01-3-arithmetic-sequences-and-series-web-images/chart%2026.tif" alt="chart%2026.tif" />
</div>
<div class="Basic-Text-Frame">
<p class="Basic-Paragraph">
<span class="char-style-override-3">(continued on next page)</span>
</p>
</div>
<div class="Basic-Text-Frame">
<p class="Problems---Problem-Part para-style-override-3">
<span>8 10</span>
</p>
</div>
<div class="Basic-Text-Frame">
<p class="Problems---Problem-Part para-style-override-3">
3
</p>
</div>
<div class="story">
<p class="Problems---EID-Section-title">
Exploring<br />
in<br />
Depth
</p>
</div>
</div>
</body>
</html>
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