I would like to know if $\displaystyle \sum_{i=2}^{n-1} i^2 = \sum_{i=2}^{n} (i-1)^2$
I feel my way of re-indexing makes sense, however, the computer shows me two different results.
The answer for $\displaystyle \sum_{i=3}^{n-1} i^2 = \frac{1}{6}(2n^3-3n^2+n-30)$. However, for $\displaystyle \sum_{i=3}^{n} (i-1)^2$, it euqals to $\displaystyle \frac{1}{6}(2n^3-3n^2+n-6)$
So if what I do is right, why there are two different answers? If it is wrong, where have I made a mistake? And what should I do?
PS: I am trying to do the reindexing because I want to use existing formula to facilitate my calculation (as required on the exam): $\displaystyle \sum_{i=1}^ni^2 = \frac{n(n+1)(2n+1)}{6}$
What would be correct re-indexing is $\sum_{i=2}^{n-1}i^2=\sum_{i=3}^n(i-1)^2$ or $\sum_{i=1}^{n-1}i^2=\sum_{i=2}^n(i-1)^2$.