Find a closed form to $\sum_{k=0}^n 2^k3^{n-k}$

Question

How would I find the closed form of

$\sum_{k=0}^n 2^k3^{n-k}$

There's two properties of summations that I think apply here:

$\sum_{k=1}^n a_kb^{i+k}=b^{i}\sum_{k=1}^n a_kb^{k}$

and

$\sum_{k=0}^n a^k=\frac{a^{n+1}-1}{a-1}$

How do I apply them?

Answer 1

Hint:

You can use the high-school formula: $$a^n-b^n=(a-b)\biggl( \sum_{k=0}^{n-1}a^{n-1-k}b^k\biggr)$$