Question

What is the formula for the general term `a_n` of 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:

Given:

`-4, 8, -14, 22, -32,...`

Note that the signs are alternating, so let's discard them and restore them at the end:

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

Write down the sequence of differences between consecutive terms:

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

Write down the sequence of differences of those differences:

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

Having reached a constant sequence, we can use the initial term of each of these sequences as coefficients for a formula to match the sequence `4, 8, 14, 22, 32`, namely:

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

`color(white)(b_n) = 4+4n-4+n^2-3n+2`

`color(white)(b_n) = n^2+n+2`

Then to get our alternating signs back, just multiply by `(-1)^n` to get:

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