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I am in advance sorry if this question is too easy for this site, but I am having real problem understanding how to solve this summation:

$$\sum_{i=1}^n{i*2^i}$$

I understand basics of summations but i don't know where to start, please help.

Filip
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1 Answers1

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Suppose you write $$\sum_{i=1}^n{i*2^i}$$ as $$x\sum_{i=1}^n{i*x^{i-1}}$$The term which is in the summation is the derivative of $$\sum_{i=1}^n{x^{i}}=\frac{x \left(x^n-1\right)}{x-1}$$ which is a geometric progression. So, compute the derivative of the rhs with respect to $x$, multiply the result by $x$ and replace $x$ by $2$.

I am sure you can take from here.

Claude Leibovici
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