I posted before several weeks this problem

Determine the following sum:

1-4+9-16+25-...-10000+10201

i got to solutions from the forum members

1 - $\displaystyle S \;=\;\sum^{101}_{k=1}k^2 - 2\sum^{50}_{k=1}(2k)^2 \;=\;\sum^{101}_{k=1}k^2 - 8\sum^{50}_{k=1}k^2$

2- $\displaystyle \sum_{n=1}^{51}\{(2n-1)^2\} - \sum_{i=1}^{50} \{(2i)^2\} $

but they give different answers

i want to know how to get the sigma notation

is it by guessing or what?