Solve $a_1^2 + 2a_2^2 + 3a_3^2 + ... + na_n^2 = a_{n+1}^2$ for natural numbers Solve the equation $\sum_{k=1}^nk\cdot a_k^2 = a_{n+1}^2$ in natural numbers (i.e. $a_n \in \mathbb N~~ \forall n \in \mathbb N$).
I really have no idea as to how to approach the issue. A starter hint would be nice! Thanks in advance!

A second general solution occurs when $a_k=k\cdot A$
$$\sum_{k=1}^n k(k\cdot A)^2=A^2\sum_{k=1}^n k^3=A^2\frac{n^2(n+1)^2}{4}$$
The final term is a perfect square for all values of $n$.
In this case, $a_{n+1}=A\frac{n(n+1)}{2}$
