Question

How do you find the explicit formula for the following sequence 2, -4, -16, -36......?

Answer 1

Examine sequences of differences to derive a polynomial formula for the sequence:

`a_n = -1/3n^3-n^2-2/3n+4`

Explanation:

A finite number of terms does not determine a unique explicit formula, but we can find a formula that will work.

Start with the given sequence:

`color(blue)(2), -4, -16, -36`

Form the sequence of differences of that sequence:

`color(blue)(-6), -12, -20`

Form the sequence of differences of that sequence:

`color(blue)(-6), -8`

Form the sequence of differences of that sequence:

`color(blue)(-2)`

We can construct a polynomial formula for `a_n` from the initial terms of these sequences:

`a_n = color(blue)(2)/(0!)+color(blue)(-6)/(1!)(n-1)+color(blue)(-6)/(2!)(n-1)(n-2)+color(blue)(-2)/(3!)(n-1)(n-2)(n-3)`

`=2-6n+6-3n^2+9n-6-1/3n^3+2n^2-11/3n+2`

`=-1/3n^3-n^2-2/3n+4`