Find closed formula by changing order of summation: $\sum_{i=1}^ni3^i$ 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?
 A: 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.
A: Just use a sum of the geometric series.
It's
$$\sum_{i=1}^n\frac{3^k(3^{n-k+1}-1)}{3-1}=\frac{n3^{n+1}}{2}-\frac{1}{2}\frac{3(3^n-1)}{2}=\frac{2n3^{n+1}-3^{n+1}+3}{4}.$$
A: 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
$$S - 3S = (3^1 + 3^2 + 3^3 + \cdots + 3^n) - n.3^{n+1}$$
$$\implies -2S = 3\left(\frac{3^n-1}{2}\right) - n.3^{n+1} = \frac{(1-2n)3^{n+1}-3}{2}$$
$$\implies S = \frac{(2n-1)3^{n+1}+3}{4}$$
A: 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.
