Question

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 $an +
b.$

Answer 1

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.