Title4

testing

<p>A generalization for all even numbers is  <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-1-Frame"><nobr><span class="math" id="MathJax-Span-1" style="width: 1.143em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.995em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.639em 1000.002em 2.631em -0.394em); top: -2.478em; left: 0.002em;"><span class="mrow" id="MathJax-Span-2"><span class="mn" id="MathJax-Span-3" style="font-family: STIXGeneral-Regular;">2</span><span class="mi" id="MathJax-Span-4" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.892em; vertical-align: -0.053em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">
2k
</script></p>

<p>where k is an integer. Therefore, a generalization for all odd numbers would be. </p>



<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-2-Frame"><nobr><span class="math" id="MathJax-Span-5" style="width: 2.979em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.631em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.639em 1000.002em 2.681em -0.394em); top: -2.478em; left: 0.002em;"><span class="mrow" id="MathJax-Span-6"><span class="mn" id="MathJax-Span-7" style="font-family: STIXGeneral-Regular;">2</span><span class="mi" id="MathJax-Span-8" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-9" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-10" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.947em; vertical-align: -0.108em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">
2k+1
</script></p>

<p>To prove that squaring an odd integer is odd, let’s square our expression. </p>



<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-3-Frame"><nobr><span class="math" id="MathJax-Span-11" style="width: 11.957em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.667em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.243em 1000.002em 2.979em -0.295em); top: -2.478em; left: 0.002em;"><span class="mrow" id="MathJax-Span-12"><span class="msubsup" id="MathJax-Span-13"><span style="display: inline-block; position: relative; width: 4.07em; height: 0px;"><span style="position: absolute; clip: rect(1.887em 1000.002em 3.375em -0.295em); top: -2.875em; left: 0.002em;"><span class="mrow" id="MathJax-Span-14"><span class="mo" id="MathJax-Span-15" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mn" id="MathJax-Span-16" style="font-family: STIXGeneral-Regular;">2</span><span class="mi" id="MathJax-Span-17" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-18" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-19" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span><span class="mo" id="MathJax-Span-20" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span><span style="display: inline-block; width: 0px; height: 2.879em;"></span></span><span style="position: absolute; top: -2.875em; left: 3.574em;"><span class="texatom" id="MathJax-Span-21"><span class="mrow" id="MathJax-Span-22"><span class="mn" id="MathJax-Span-23" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span></span></span><span style="display: inline-block; width: 0px; height: 2.284em;"></span></span></span></span><span class="mo" id="MathJax-Span-24" style="font-family: STIXGeneral-Regular; padding-left: 0.3em;">=</span><span class="mn" id="MathJax-Span-25" style="font-family: STIXGeneral-Regular; padding-left: 0.3em;">4</span><span class="msubsup" id="MathJax-Span-26"><span style="display: inline-block; position: relative; width: 0.995em; height: 0px;"><span style="position: absolute; clip: rect(1.639em 1000.002em 2.631em -0.444em); top: -2.478em; left: 0.002em;"><span class="mi" id="MathJax-Span-27" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span><span style="position: absolute; top: -2.676em; left: 0.499em;"><span class="mn" id="MathJax-Span-28" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 2.284em;"></span></span></span></span><span class="mo" id="MathJax-Span-29" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-30" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">4</span><span class="mi" id="MathJax-Span-31" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-32" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-33" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.725em; vertical-align: -0.442em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">
\left( 2k + 1 \right)^{2} = 4k^2 + 4k + 1
</script></p>

<p>Let’s factor out <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-4-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-34" style="width: 1.143em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.995em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.639em 1000.002em 2.631em -0.444em); top: -2.478em; left: 0.002em;"><span class="mrow" id="MathJax-Span-35"><span class="mn" id="MathJax-Span-36" style="font-family: STIXGeneral-Regular;">4</span><span class="mi" id="MathJax-Span-37" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.892em; vertical-align: -0.053em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-4">4k</script></p>



<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-5-Frame"><nobr><span class="math" id="MathJax-Span-38" style="width: 6.649em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.905em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.887em 1000.002em 3.375em -0.444em); top: -2.875em; left: 0.002em;"><span class="mrow" id="MathJax-Span-39"><span class="mn" id="MathJax-Span-40" style="font-family: STIXGeneral-Regular;">4</span><span class="mi" id="MathJax-Span-41" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mrow" id="MathJax-Span-42" style="padding-left: 0.201em;"><span class="mo" id="MathJax-Span-43" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mi" id="MathJax-Span-44" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-45" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-46" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span><span class="mo" id="MathJax-Span-47" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span><span class="mo" id="MathJax-Span-48" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-49" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.879em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.503em; vertical-align: -0.442em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">
4k\left(k+1\right) + 1
</script></p>



<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-6-Frame"><nobr><span class="math" id="MathJax-Span-50" style="width: 7.889em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.046em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.887em 1000.002em 3.375em -0.394em); top: -2.875em; left: 0.002em;"><span class="mrow" id="MathJax-Span-51"><span class="mn" id="MathJax-Span-52" style="font-family: STIXGeneral-Regular;">2</span><span class="mrow" id="MathJax-Span-53" style="padding-left: 0.201em;"><span class="mo" id="MathJax-Span-54" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mi" id="MathJax-Span-55" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mrow" id="MathJax-Span-56" style="padding-left: 0.201em;"><span class="mo" id="MathJax-Span-57" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mi" id="MathJax-Span-58" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-59" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-60" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span><span class="mo" id="MathJax-Span-61" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span><span class="mo" id="MathJax-Span-62" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span><span class="mo" id="MathJax-Span-63" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-64" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.879em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.503em; vertical-align: -0.442em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">
2\left(k\left(k+1\right)\right) + 1
</script></p>

<p>Since <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-7-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-65" style="width: 5.26em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.665em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.887em 1000.002em 3.375em -0.295em); top: -2.875em; left: 0.002em;"><span class="mrow" id="MathJax-Span-66"><span class="mrow" id="MathJax-Span-67"><span class="mo" id="MathJax-Span-68" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mi" id="MathJax-Span-69" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mrow" id="MathJax-Span-70" style="padding-left: 0.201em;"><span class="mo" id="MathJax-Span-71" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mi" id="MathJax-Span-72" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-73" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-74" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span><span class="mo" id="MathJax-Span-75" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span><span class="mo" id="MathJax-Span-76" style="vertical-align: -0.196em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.879em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.503em; vertical-align: -0.442em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-7">\left(k\left(k+1\right)\right)</script> is still an integer, and we’re working with generalizations, we can just merge that back into a generic constant <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-8-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-77" style="width: 0.499em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.449em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.639em 1000.002em 2.631em -0.444em); top: -2.478em; left: 0.002em;"><span class="mrow" id="MathJax-Span-78"><span class="mi" id="MathJax-Span-79" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.892em; vertical-align: -0.053em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-8">k</script>, and we’re back to our original odd-form. </p>



<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-9-Frame"><nobr><span class="math" id="MathJax-Span-80" style="width: 2.979em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.631em; height: 0px; font-size: 112%;"><span style="position: absolute; clip: rect(1.639em 1000.002em 2.681em -0.394em); top: -2.478em; left: 0.002em;"><span class="mrow" id="MathJax-Span-81"><span class="mn" id="MathJax-Span-82" style="font-family: STIXGeneral-Regular;">2</span><span class="mi" id="MathJax-Span-83" style="font-family: STIXGeneral-Italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-84" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">+</span><span class="mn" id="MathJax-Span-85" style="font-family: STIXGeneral-Regular; padding-left: 0.25em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.483em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.947em; vertical-align: -0.108em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-9">
2k + 1
</script></p>