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I dont understand how to solve this question:

I need to find a closed formula for :

$$\sum_{k=0}^n k\,5^k$$

Thanks a lot!

lulu
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Ron Swanson
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2 Answers2

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Here is an elementary approach, without calculus:

$$\begin{align*} \sum_{k=0}^nk5^k&=\sum_{k=1}^nk5^k\\ &\overset{(1)}=\sum_{k=1}^n\sum_{\ell=1}^k5^k\\ &\overset{(2)}=\sum_{\ell=1}^n\sum_{k=\ell}^n5^k\\ &\overset{(3)}=\sum_{\ell=1}^n\frac{5^{n+1}-5^\ell}{5-1}\\ &=\frac14\sum_{\ell=1}^n\left(5^{n+1}-5^\ell\right)\\ &=\frac14\left(\sum_{\ell=1}^n5^{n+1}-\sum_{\ell=1}^n5^\ell\right)\\ &=\frac14\left(n5^{n+1}-\frac{5^{n+1}-5}{5-1}\right)\\ &=\frac14\left(n5^{n+1}-\frac14\left(5^{n+1}-5\right)\right)\\ &=\frac1{16}\left(4n5^{n+1}-5^{n+1}+5\right)\\ &=\frac5{16}\big((4n-1)5^n+1\big) \end{align*}$$

$(1)$: The $k=0$ term is $0$ anyway.
$(2)$: Reverse the order of summation.
$(3)$: Sum of a finite geometric series.

The basic idea here is the same as in this argument for the sum of a similar infinite series.

Community
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Brian M. Scott
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Hint: Generalise the question and find a closed form for $$\sum_{k=0}^n kx^k=x\sum_{k=1}^{n-1} kx^{k-1}=x\biggl(\sum_{k=0}^{n-1}x^k\biggr)'.$$

Bernard
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  • This hint seems to duplicate some of the answers in this faq version of the question. –  Jul 02 '16 at 21:36
  • @T.Bongers: For many students there is a significant difference between the finite and the infinite sum, the former being the more difficult. – Brian M. Scott Jul 02 '16 at 21:38
  • @BrianM.Scott Indeed, and at least two of the answers in the post I linked start by computing the explicit closed form for the finite sum. Rather than fragmenting answers over the dozens of times that this question gets asked, I feel it's better to contribute good answers to a single thread. –  Jul 02 '16 at 21:39
  • @Brian M. Scott: The formula is harder to remember in the finite case, but it's in the curriculum of high school, which the infinite case is not. – Bernard Jul 02 '16 at 21:43
  • @T.Bongers: Which is fine, if the reader doesn’t get lost in the clutter. It’s fine to refer students to that question, since it does have a nice variety of answers, but it’s also a bit overwhelming, and I do not consider this a duplicate of it. – Brian M. Scott Jul 02 '16 at 21:44
  • @Bernard: In some $40$ years of teaching I don’t think that I ever had a student who had seen the finite case in high school. The finite case of a geometric series, sure, but not this. – Brian M. Scott Jul 02 '16 at 21:44
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    @Brian M. Scott: In France, they have, under the name ‘sum of consecutive terms of a geometric sequence’, and they have to use it in probability. – Bernard Jul 02 '16 at 21:47
  • @Bernard: I’m not altogether surprised; the French and U.S. systems have very different curricula. – Brian M. Scott Jul 02 '16 at 21:49
  • @Bernard I think it should be $$=x\biggl(\sum_{k=1}^{n-1}x^k\biggr)'$$ – Amirali Mar 30 '21 at 19:55
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    @amirali: That's right. I guess I was thinking of the final formula – which I didn't write, and resulted in another of my lapsus calami… 'Tis fixed now. Thank you for warning me! – Bernard Mar 30 '21 at 20:04