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Working on homework for a probability and computing class, but my ability to work with summations is rusty to say the least, so I suspect this is going to turn out pretty straightforward.

Problem asks to find a closed formula for $$\sum_{i=1}^ni3^i$$ by representing it as a double sum and changing the order of summation. I did that by following a hint from the instructor and came up with $$\sum_{k=1}^n\sum_{i=k}^n3^i,$$ but I'm not really sure what that accomplished. What's the next step? What am I looking for here?

Michael Rozenberg
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lordgankoo
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  • 1
    Alternative: integrate with respect to 3 – Holographer Sep 04 '17 at 22:39
  • You can also have look at some similar sums and generalizations: https://math.stackexchange.com/questions/11464/how-to-compute-the-formula-sum-limits-r-1d-r-cdot-2r https://math.stackexchange.com/questions/87030/proving-sum-limits-i-0n-i-2i-1-n1-2n-1-by-induction https://math.stackexchange.com/questions/90637/sum-limits-n-1-inftyn-frac23n-evalute-sum https://math.stackexchange.com/questions/30732/how-can-i-evaluate-sum-n-0-infty-n1xn – Martin Sleziak Sep 05 '17 at 05:12
  • And you can find a few other posts about similar sums using Approach0. – Martin Sleziak Sep 05 '17 at 05:17

4 Answers4

10

Here is a rather detailed elaboration which might be useful.

We obtain \begin{align*} \color{blue}{\sum_{i=1}^ni3^i}&=\sum_{i=1}^n\left(\sum_{k=1}^i 1\right)3^i\tag{1}\\ &=\sum_{i=1}^n\sum_{k=1}^i 3^i =\sum_{1\leq k\leq i\leq n}3^i =\sum_{k=1}^n\sum_{i=k}^n3^i\tag{2}\\ &=\sum_{k=1}^n\sum_{i=0}^{n-k}3^{i+k}\tag{3}\\ &=\sum_{k=1}^n3^k\cdot\frac{3^{n-k+1}-1}{3-1}\tag{4}\\ &=\frac{1}{2}\sum_{k=1}^n\left(3^{n+1}-3^k\right)\tag{5}\\ &=\frac{n}{2}3^{n+1}-\frac{1}{2}\sum_{k=1}^n3^k\tag{6}\\ &=\frac{n}{2}3^{n+1}-\frac{1}{2}\cdot\left(\frac{3^{n+1}-1}{3-1}-1\right)\tag{7}\\ &=\frac{n}{2}3^{n+1}-\frac{1}{4}3^{n+1}+\frac{3}{4}\tag{8}\\ &\color{blue}{=\frac{n}{4}(2n-1)3^{n+1}+\frac{3}{4}}\tag{9} \end{align*}

Comment:

  • In (1) we represent the factor $i$ as sum.

  • In (2) we multiply out in the left-hand sum and write the index range somewhat more conveniently in the middle sum. We exchange the sums in the right-hand double-sum.

  • In (3) we shift the index of the inner sum to start from $i=0$.

  • In (4) we apply the finite geometric summation formula.

  • In (5) we do some simplifications.

  • In (6) we multiply out and do some simplifications.

  • In (7) we apply the finite geometric summation formula again.

  • In (8) and (9) we do some more simplifications.

Markus Scheuer
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  • Looks like I'm really gonna need to do some review on the geometric summation formula, but otherwise this was pretty helpful. Thanks! – lordgankoo Sep 04 '17 at 20:24
  • @lordgankoo: You're welcome! :-) I've adjusted the expression in (7) somewhat to better see the connection with the formula in (4). Regards, – Markus Scheuer Sep 04 '17 at 20:30
3

Just use a sum of the geometric series.

It's $$\sum_{i=1}^n\frac{3^k(3^{n-k+1}-1)}{3-1}=\frac{n3^{n+1}}{2}-\frac{1}{2}\frac{3(3^n-1)}{2}=\frac{2n3^{n+1}-3^{n+1}+3}{4}.$$

Michael Rozenberg
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3

Because the other answer already discusses about computing $\sum_{k=1}^n \sum_{i=k}^n 3^i$, here's another way to compute $\sum_{i=1}^n i3^i$

\begin{eqnarray} S & = & \sum_{i=1}^n i 3^i \\ & = & 1.3^1 + 2.3^2 + 3.3^3 + \cdots + n.3^n \\ 3S & = & \space\space\space\space\space\space\space\space\space\space\space\space 1.3^2 + 2.3^3 + \cdots + (n-1).3^n + n.3^{n+1} \\ \end{eqnarray}

Therefore, on subtracting, we get $$S - 3S = (3^1 + 3^2 + 3^3 + \cdots + 3^n) - n.3^{n+1}$$ $$\implies -2S = 3\left(\frac{3^n-1}{2}\right) - n.3^{n+1} = \frac{(1-2n)3^{n+1}-3}{2}$$ $$\implies S = \frac{(2n-1)3^{n+1}+3}{4}$$

kishlaya
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2

The inner sum is a geometric series. It can be expanded as follows:

$$\sum_{k=1}^n\sum_{i=k}^n3^i=\sum_{k=1}^n(\sum_{i=k}^n3^i)=\sum_{k=1}^n\frac{3^k-3^{n+1}}{1-3}=\sum_{k=1}^n\frac{3^{n+1}-3^k}{2}=\frac{n3^{n+1}}2-\frac12\sum_{k=1}^n3^k$$

The rest should be pretty straight-forward.

Tyler Borgard
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