How do i simplify this in terms of n?
$\displaystyle
\sum_{i=0}^{n-1}{2^{i} (n-i)}
$
Cheers!
$\displaystyle \sum_{i=0}^{n-1}{2^{i} (n-i)}=\frac{1}{2}\sum_{i=1}^{n}2^{i}+\frac{n}{2}\sum _{i=1}^{n}2^{i}-\frac{1}{2}\sum_{i=1}^{n}i2^{i}$
Try using the partial sum for a geometric series formula.
$\displaystyle S_{n}=a_{n}\left(\frac{1-r^{n}}{1-r}\right)$, where
r is the common ratio and a is the first term.
So, from the first one we have $\displaystyle \frac{1-2^{n}}{1-2}=2^{n}-1$
The second one is the same only multiplied by n. Giving $\displaystyle n(2^{n}-1)$
The third only can be found by differentiating the partial sum formula.
$\displaystyle \sum_{i=1}^{n}x^{i}=\frac{x(1-x^{n})}{1-x}$
$\displaystyle \frac{d}{dx}\left[\sum_{i=1}^{n}x^{i}\right]=\sum_{i=1}^{n}ix^{i-1}=\frac{(nx-n-1)x^{n}+1}{(x-1)^{2}}$
$\displaystyle \sum_{i=1}^{n}ix^{i}=\frac{x((nx-x-1)x^{n}+1)}{(x-1)^{2}}$
Sub in x=2 and we get $\displaystyle 2(2^{n}(n-1)+1)$.
Don't forget to divide by 2 because of the 1/2.
$\displaystyle \frac{1}{2}\sum_{i=1}^{n}i2^{i}=(n-1)2^{n}+1$
Add this with the other two sums and we get:
$\displaystyle 2^{n}-1+n(2^{n}-1)-((n-1)2^{n}+1)=\boxed{2^{n+1}-n-2}$
Just one way to go about it. Remember, you can play around with the geometric series formulas by differentiating, integrating, and so on, and get away with a lot.
Here's another way. Let
$\displaystyle
S_n = \sum_{i=0}^{n-1} 2^i(n-i) \;\;\;(1)
$
multiplying by 2 gives
$\displaystyle
2S_n = \sum_{i=0}^{n-1} 2^{i+1}(n-i).
$
Now shift
$\displaystyle
2S_n = \sum_{i=1}^{n} 2^i(n-i+1)\;\;\;(2).
$
Now substract (1) from (2) so
$\displaystyle
S_n = \sum_{i=1}^{n} 2^i(n-i+1) - \sum_{i=0}^{n-1} 2^i(n-i)
$
$\displaystyle
= \sum_{i=1}^n 2^i - n = 2^{n+1} - n - 2.
$