Finding generating function in a closed form : $\sum_{n=1}^N(3n^2-10n)\cdot2^n$

Question

I'm currently stuck on the last section of my assignment...

It asks me to find a generating function in a closed form

the only part im struggling is $\sum_{n=1}^N(3n^2-10n)\cdot2^n$ or $\sum_{n=1}^N(\text{quadratic}(n))C^n$

I have no clue on how to convert this...

any assistance would be appreciated thank you

Answer 1

Hint. Here is a general route.

We assume $x \ne 1$. One may recall the standard geometric evaluation: $$ 1+x+x^2+...+x^n=\frac{1-x^{n+1}}{1-x},\tag1 $$ then by differentiating $(1)$ we have $$ 1+2x+3x^2+...+nx^{n-1}=\frac{1-x^{n+1}}{(1-x)^2}-\frac{(n+1)x^{n}}{1-x}, \tag2 $$ multiplying $(2)$ by $x$ and differentiating once more gives $$ 1+2^2x+3^2x^2+...+n^2x^{n-1}=\frac{d}{dx}\left(\frac{1-x^{n+1}}{(1-x)^2}-\frac{(n+1)x^{n}}{1-x}\right). \tag3 $$