# calculate the sum of n initial terms of the sequence

I have problem with finding the formula for the sum of $n$ initial terms of $$a(n)=n \cdot 2^n.$$

I only wrote the first few terms $\displaystyle a(1)=2$, $a(2)=8$, $a(3)=24$, $a(4)=64$

-
I don't understand, what are you asking for? You already wrote out the first couple of terms, and you have the generic form of an $n$-th term. Do you want to sum the first $n$ terms? –  gt6989b Feb 18 '14 at 17:14
yes, that's what I am asking for –  Gregor Feb 18 '14 at 17:16

Note that if we take the usual geometric series $$\sum_{k=0}^n x^k = \frac{1 - x^{k+1}}{1-x}, \quad \forall x \neq 0,$$ we can differentiate it with respect to $x$ to get $$\frac{d}{dx} \sum_{k=0}^n x^k = \sum_{k=1}^n k x^{k-1} = \frac{1}{x} \sum_{k=1}^n k x^k$$ on the left-hand side. Now differentiate the right-hand side using the quotient rule and you should be able to take it from there.