Question

Given 10, 17, 24, 31, the nth term is given by 7n+3. How you know?

Answer 1

By realizing that `10 = 7(1) + 3`, `17 = 7(2) + 3`, etc.

An arithmetic sequence is one where some constant difference between numbers in a sequence gives you a pattern to get to the next unlisted number in the sequence.

In this case, the difference is `7`. When one works backwards and subtracts `7` from `10`, the result is `3`. At the `0`th term, we assume that `n > 0` for series and sequences.

So, `3` is the `0`th term, giving us the y-intercept, so to speak. Now we would simply have to recognize that if we increment `n` integrally, and we have a difference of `7` from term to term, that is the operation `7n`.

So, the arithmetic sequence would be generated as:

`color(green)(a_0) = 3 + 7(0) = color(green)(3)`

`color(green)(a_(1)) = a_0 + 7(1) = color(green)(10)`

`color(green)(a_(2)) = a_(1) + 7(1) = a_0 + 7(1) + 7(1) = color(green)(17)`

`color(green)(a_(3)) = a_(2) + 7(1) = a_1 + 7(1) + 7(1)`

`= a_0 + 7(1) + 7(1) + 7(1) = color(green)(24)`

and so on. Each term has `a_0` within it, added to `7` times the current index indicated by `a_n`. So, we just have the recursive definition:

`color(blue)(a_n = a_(n-1) + 7)`

or the overall definition:

`color(blue)(a_n = 3 + 7n)`