Combinatorial proof of the identity:

Question

$$\sum_{k=1}^n {(k^3)} = \left(\sum_{k=1}^n {k}\right)^2 $$

How could I approach this problem to make a combinatorial proof?

Answer 1

Try induction. Suppose the statement holds for $n-1$. To prove it for $n$, write: $$ \left( \sum_{k=1}^{n} k \right)^2 = \left( n + \sum_{k=1}^{n-1} k \right)^2 = n^2 + 2 n \sum_{k=1}^{n-1} k + \left( \sum_{k=1}^{n-1} k \right)^2 = n^2 + 2 n \sum_{k=1}^{n-1} k + \sum_{k=1}^{n-1} k^3. $$ Now use $$ \sum_{k=1}^{n-1} k = {(n-1) \, n \over 2}. $$