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

Feb 14, 2016

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

${a}_{n} = - \frac{1}{3} {n}^{3} - {n}^{2} - \frac{2}{3} n + 4$

Explanation:

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

$\textcolor{b l u e}{2} , - 4 , - 16 , - 36$

Form the sequence of differences of that sequence:

$\textcolor{b l u e}{- 6} , - 12 , - 20$

Form the sequence of differences of that sequence:

$\textcolor{b l u e}{- 6} , - 8$

Form the sequence of differences of that sequence:

$\textcolor{b l u e}{- 2}$

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)

$= 2 - 6 n + 6 - 3 {n}^{2} + 9 n - 6 - \frac{1}{3} {n}^{3} + 2 {n}^{2} - \frac{11}{3} n + 2$

$= - \frac{1}{3} {n}^{3} - {n}^{2} - \frac{2}{3} n + 4$