Summation of $\sum_{i=1}^n i2^i.$

Question

An intermediate step in a problem I was working on was to find a closed form for the sum $$\sum_{i=1}^n i2^i.$$

WolframAlpha returns $2^{n+1}(n-1) + 2$, but didn't provide any step-by-step solution. I would usually provide progress that I made, but I wasn't sure how to proceed(although I solved the problem by bashing out small values).

Answer 1

To find the formula on your own. Write $$ S = \sum_{i=1}^n i2^i. \tag{1}$$ Then $$ 2S = \sum_{i=1}^n i2^{i+1} = \sum_{i=2}^{n+1} (i-1)2^i \tag{2}$$ Subtract (1) from (2) $$ S = 2S - S = \sum_{i=2}^{n+1} (i-1)2^i - \sum_{i=1}^n i2^i \\ = n2^{n+1} +\sum_{i=2}^n (-1)2^i -1\cdot 2^1 $$ The sum remaining is a geometric series, I assume you already know how to do it $$ S = n2^{n+1} - \left(\frac{2^{n+1}-2^2}{2-1}\right) - 2= (n-1)2^{n+1} + 2 $$

Answer 2

Let the sum equals $S$.$$\big(1\cdot x+2\cdot x^2+\cdots +n\cdot x^n\big)+\big(n\cdot x^n+(n-1)x^{n-1}+\cdots +x\big)=(n+1)(1+x+\cdots+x^n-1)$$ Hence, $2S=(n+1)(1+x+\cdots +x^n)-(n+1)=(n+1)\cdot \frac{x^{n+1}-1}{x-1}-(n+1) $ and finally we get $$S=\frac{n+1}{2}(\frac{x^{n+1}-1}{x-1})-\frac{n+1}{2}$$ If you put $x=2$ you will get the required sum.

Answer 3

I assume you know the formula for the geometric sum:

$$\sum_{k=0}^n x^k=\frac{x^{n+1}-1}{x-1}$$

It can be shown by multiplication by $x-1$ and cancellation of terms.

Your sum is a little more complicated.

$$\sum_{k=1}^n kx^k$$

However, we can write it as a triangular array. Let's use $n=5$ for clarity.

$$\begin{array}( x^5 & + & x^4 & + & x^3 & + & x^2 & + & x \\ x^5 & + & x^4 & + & x^3 & + & x^2 \\ x^5 & + & x^4 & + & x^3 \\ x^5 & + & x^4 \\ x^5 \end{array}$$

Can you see the general pattern? Maybe this will help.

Answer 4

Hint:

$$2S_4-S_4=(1\cdot2^2+2\cdot2^3+3\cdot2^4+4\cdot2^5)-(1\cdot2^1+2\cdot2^2+3\cdot2^3+4\cdot2^4) \\=4\cdot2^5-(2^1+2^2+2^3+2^4)=3\cdot2^5+2.$$

You can easily generalize.