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Formula for calculating $\sum_{n=0}^{m}nr^n$

Can someone explain step by step how to derive the following identity? $$\sum_{k=1}^{n} k \cdot 2^k = 2(n \cdot 2^n - 2^n + 1) $$

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marked as duplicate by lhf, Kannappan Sampath, Asaf Karagila, Hans Lundmark, t.b. Mar 15 '12 at 10:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

What formulas related to this one do you know? – Davide Giraudo Mar 14 '12 at 15:39
Related question – Tom Cooney Mar 14 '12 at 15:42
I recall the basic ones, that we are expected to know: sum of k = 1 to n of k is n(n+1) / 2 and geometric series formula ie. sum of n = 0 to m of a* r ^n – rrazd Mar 14 '12 at 15:42
Isn't this a multi duplicate? – Did Mar 14 '12 at 23:21

3 Answers 3

up vote 9 down vote accepted

If you know $\sum_{k=1}^n a^k=\frac{a^{n+1}-1}{a-1}$, take the derivative of both sides with respect to $a$, multiply by $a$, and set $a=2$

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Induction is a surefire way to prove it with elementary methods. With calculus and some formulas, we can approach this problem by evaluating a derivative in two different ways, as follows:

Quotient rule: $$\rm \frac{d}{dx}\frac{x^n-1}{x-1} =\frac{n\,x^{n-1}(x-1)-(x^n-1)(1)}{(x-1)^2} \tag{1}$$

Geometric formula:

$$\rm \frac{d}{dx}\frac{x^n-1}{x-1}=0+1x^0+2x^1+\cdots+n\,x^{n-1} \tag{2}$$

Equate equation $(1)$ with equation $(2)$, multiply both sides by $\rm x$ and then set $\rm x=2$.

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Your last term in $(2)$ should be $nx^{n-1}$ instead – Kirthi Raman Mar 14 '12 at 16:07

Although you got great answers, I will add another way to look at the expression you are looking for

Let us denote

$$ S = \sum_{k=1}^{n} k \cdot 2^k $$

This can be expanded as

$$ \begin{align*} S &= 2 + 2 . 2^2 + 3 . 2^3 + \cdots n 2^n\\ 2S &= \hspace{8pt}+1.2^2+2.2^3 + \cdots (n-1) . 2^n+n . 2^{n+1} \end{align*} $$

If you notice, by multiplying by $2$ and writing under similar terms and then subtracting, we get

$$ \begin{align*} S = n . 2^{n+1} - \left(2+2^2+2^3+\cdots+2^n \right) &= n .2^{n+1}-2^{n+1}+2\\ &= 2(n . 2^n-2^n+1) \end{align*} $$

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