Question

For the series `k_n=2^n+1`, what is `k`? what is `n`? what is the value of `k` when `n=5`?

Answer 1

`n` is the position of the term within the sequence
`k` is the value of the `n`-th term

The 5th number in the sequence is `33`.

Explanation:

In this problem, `k` represents the `n`-th term in the sequence.

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When `n= color(red)1`:

`k = 2^color(red)1+1 = 2+1 = color(blue)3`

When `n= color(red)2`:

`k = 2^color(red)2+1 = 4+1 = color(blue)5`

When `n= color(red)3`:

`k = 2^color(red)3+1 = 8+1 = color(blue)9`

When `n= color(red)4`:

`k = 2^color(red)4+1 = 16+1 = color(blue)17`

This is the sequence that the problem gives us: `color(blue)3, color(blue)5, color(blue)9, color(blue)17...`

Our job now is to find the next term in the sequence.

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We can do this by plugging in `n=color(red)5`, so that `k` will be the `5`th term in the sequence.

`k = 2^color(red)5 + 1 = 32 + 1 = color(blue)33`

Final Answer

Answer 2

`k` is the actual number;
`n` is the position of that number in the sequence.

`k_5=2^5+1=32+1=33`

Explanation:

[Note that I rewrote your question to use `k_n` to denote the `n`th value for `k` in the sequence (and to add some clarity)]

`k_5` is the `5`th number in the sequence