Prove by Induction with Summation Struggling with this , especially with the double summation; if anyone can help it be much appreciated!
$$\forall n \in \Bbb N : \quad \sum_{i=1}^n \sum_{c=1}^i c = \sum_{i=1}^n i(n − i + 1) .$$
It needs to be answered in the following format:
1- Prove for n=1
2- assume n=k
3- prove for k+1

Rewrite the identity for $n+1$:


*

*Left hand side
$$\sum_{i=1}^{n+1} \sum_{c=1}^i c =\sum_{i=1}^{n} \sum_{c=1}^i c+\sum_{c=1}^{n+1} c.$$

*Right hand side
$$\sum_{i=1}^{n+1} i(n+1 − i + 1)=\sum_{i=1}^{n+1} i(n− i + 1)+\sum_{i=1}^{n+1}i=\sum_{i=1}^{n} i(n− i + 1)+\sum_{i=1}^{n+1}i.$$
As the additional terms are the same, if the identity holds for $n$, then it holds for $n+1$.
