Question

What is the formula for the sequence: `-4, 8, -14, 22, -32,...` ?

Answer 1

Separate out the `+-` signs, calculate a formula for the resulting positive sequence by looking at differences, then restore the alternating signs.

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

Explanation:

I'm sure I have answered this question before, but here's how you solve it:

First note that the signs of the terms alternate between `+` and `-`. So leave the signs out to start - we will add them again at the end:

Let `b_1, b_2, b_3, b_4, b_5` be the sequence:

`color(blue)(4), 8, 14, 22, 32`

Form the sequence of differences of that sequence to get the sequence:

`color(blue)(4), 6, 8, 10`

Form the sequence of differences of that sequence to get the sequence:

`color(blue)(2), 2, 2`

Having reached a constant sequence, the first term of each of these sequences are the coefficients of a formula for `b_n` as follows:

`b_n = color(blue)(4)/(0!) + color(blue)(4)/(1!)(n - 1) + color(blue)(2)/(2!)(n-1)(n-2)`

`=4+4n-4+n^2-3n+2 = n^2+n+2`

Then the original sequence with signs is given by the formula:

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