Question

What is the formula for the sequence `2, -1, 4, -7, 10, -13, 16,...` ?

Answer 1

The given sequence is matched by the formula:

`a_n = (-1)^n (-2+3(n-1))`

or if you prefer:

`a_n = (-1)^n (3n-5)`

Explanation:

No infinite sequence is determined purely by a finite number of terms, unless you are given further information - e.g. that the sequence is arithmetic or geometric.

That having been said, note that if we multiply the terms of the given sequence by `(-1)^n`, then we get the sequence:

`-2, 1, 4, 7, 10, 13, 16,...`

which is (as far as it goes) an arithmetic sequence with initial term `-2` and common difference `3`.

So a formula that fits the original sequence can be written:

`a_n = (-1)^n (-2+3(n-1))`