I need some advice on solving this summation

Question

Suppose we are asked to solve the following summation

$$\sum_{i=1}^{n} (i2^{i-1})$$

Below is my attempt at solving this..

I know that the quotient rule for exponents can help me simplify the summation.

$$\sum_{i=1}^{n} (i2^{i}/2)$$

now i can extract the constant from the summation..

$$ 1/2\sum_{i=1}^{n} (i2^{i})$$

this is where i am stuck.. can anyone give me some advice on what to do next?

Answer 1

$$\sum_{i=0}^ni2^{i-1}=\left(\sum_{i=0}^nx^i\right)'_{x=2}=\left(\frac{x^{n+1}-1}{x-1}\right)'_{x=2}=$$ $$=\left(\frac{(n+1)x^{n}(x-1)-x^{n+1}+1}{(x-1)^2}\right)_{x=2}=(n+1)2^{n}-2^{n+1}+1.$$

Answer 2

If you don't want to do calculus, you need to solve for it instead of directly calculating it. Define $S$ as,

$$S=\sum_{i=0}^n(i2^{i-1}),$$

$$2S=\sum_{i=0}^n(i2^i)=\sum_{i=1}^{n+1}[(i-1)2^{i-1}],$$

$$2S-S=n2^n+\sum_{i=1}^n\left\{[(i-1)-i]2^{i-1}\right\}=n2^n-\sum_{i=1}^n 2^{i-1}=n2^n-2^n+1=(n-1)2^n+1.$$