1. ## Algebraic Manipulation

I have a math problem where the expression:

$\displaystyle \displaystyle\sum_{i=0}^{n-1}(i+1)(n-i)$

is changed to:

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

but I am having difficulty understanding how this is achieved.
Can anyone explain how this is done?

Thanks

2. Originally Posted by wmassey
I have a math problem where the expression:

$\displaystyle \displaystyle\sum_{i=0}^{n-1}(i+1)(n-i)$

is changed to:

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

but I am having difficulty understanding how this is achieved.
Can anyone explain how this is done?

Thanks
In the second expression the $\displaystyle i$ is one greater than it is in the first one because it starts at 1 (not 0) and goes to n (not n-1). You could account for this by subtracting one from the $\displaystyle i$ when changing the limits of the summation:
$\displaystyle \displaystyle\sum_{i=0}^{n-1}[i+1][n-i]$
$\displaystyle \displaystyle\sum_{i=1}^{n}[(i-1)+1][n-(i-1)]$
$\displaystyle \displaystyle\sum_{i=1}^{n}i(n+1-i)$

Does that make any sense?

3. Yea I understand it now - thanks pflo!