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.

f

step-by-step explanation:

f

square pyramid

step-by-step explanation:

square prism looks like nothing like that

triangular prism has a rectangular base instead of a square base

tetrahedron has a triangular base