The closed form sum of $$12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]$$ for $n \geq 1$ is $n(n+1)(n+2)(an+b).$ find $an + b.$

n is an unspecified positive integer.  If you want a single numerical result for the sum you must specify the value for n.

For example:

n = 5    means  n(n+1)(n+2)(3n+1) = 3360

n=9     means   n(n+1)(n+2)(3n+1) = 27720

etc.

The sum is a function of n.

the answer is 510

step-by-step explanation:

15 • 13 = 195

15 • 12 = 180

15 • 5 = 75

12 • 5 = 60

195 + 180 = 375

375 + 75 = 450

450 + 60 = 510