Number theory with positive integer $n$ question If $n$ is a positive integer, what is the smallest value of $n$ such that $$(n+20)+(n+21)+(n+22)+ ... + (n+100)$$ is a perfect square?
I don't even now how to start answering this question. 

Hint: Let $$f(n) = (n+20)+(n+21)+(n+22)+ ... + (n+100)=81n+\frac{100\cdot101}{2}-\frac{19\cdot20}{2}=81n+4860=81(n+60)$$
Now since $81$ is already a perfect square you have to find the smallest $n$ for which $n+60$ is a perfect square too...$\Rightarrow n=4$ and $f(n) = 72^2$
