a[1], a[2], a[3], \cdots is an arithmetic progression. b[1], b[2], \cdots is also an arithmetic progression, such that b[i] \in \mathbb{N} for every [i]. Is a[b[1]], a[b[2]], \cdots an arithmetic progression as well?
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What are we doing here?
- Composing Sequences
- How many sequences?
- Two
- Can we compose more sequences?
- Yes, but let's try if we can do just two for now.
- Also, more than two sequences look complicated.
- Which two sequences should we compose?
- Self composition seems like an interesting idea.
- How many sequences?
- What else can we do on two sequences?
- Do some pointwise operations on them
- Add them
- Multiply them
- Compare them
- Does the comparison flip at any point in the case of Arithmetic Progressions?
- Yes, but at most once.
- Does the comparison flip at any point in the case of Arithmetic Progressions?
- Some limiting stuff
- Probably too hard for low level users
- Convolute them
- Hard to motivate them
- Do some pointwise operations on them
- Composing Sequences
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Assuming we are going with composition, what is the directive?
- Two arithmetic sequences. Is the composition also an arithmetic sequence?
- What else could we ask along similar lines?
- Ask what is the new common difference
- Ask something about parity, or some residue
- Does the composition commute?
Alice likes the natural numbers:
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \ldots ]
But he likes odd numbers more. So, he does the following
((*)) He chooses the numbers at odd positions, i.e, those at the first, third, fifth, seventh and so on positions, and removes the rest.
So, now he has
[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \ldots ]
- We are composing an arithmetic sequence with itself above.
- Maybe we can compose it multiple times.
- Should we compose it only finitely many times?
- What can we ask?
- Finite Case
- What is the new common difference after doing ((*)) a few times?
- Is a certain number removed after this process?
- When is a certain number removed?
- What is the next number after 1?
- Infinite Case
- Does anything other than other than 1 remain?
- Finite Case
But he does ((*)) again, and has
[ 1, 5, 9, 13, 17, 21, 25, 29, 33, 37 \ldots ]
Alice does ((*)) 8 more times, (and thus 10 times in total). What is the next number after 1 in the sequence that he has now?
Alice likes the natural numbers:
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \ldots ]
But he likes odd numbers more. So, he does the following
((*)) He chooses the numbers at odd positions, i.e, those at the first, third, fifth, seventh and so on positions, and removes the rest.
So, now he has
[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \ldots ]
But he does ((*)) again, and has
[ 1, 5, 9, 13, 17, 21, 25, 29, 33, 37 \ldots ]
Alice does ((*)) 8 more times, (and thus 10 times in total).
What is the next number after 1 in the sequence that he has now?
[ ] (1 + 2^{10}) [ ] (1 + 2 \times {10}) [ ] (2^{10})