Consider:

      (15 + 25)     +     (17 + 27)     =   2 *(1 + 2)4   
   (15 + 25 + 35)   +   (17 + 27 + 37)   =  2 *(1 + 2 + 3)4 
(15 + 25 + 35 + 45) + (17 + 27 + 37 + 47) = 2 *(1 + 2 + 3 + 4)4
       ...                  ...       and so on     ...

First, verify that both sides are equal for further increase in n,

then prove it.

(In reply to computer verification no proof by Charlie)

The LHS is a 7th degree polynomial in n, and the RHS is an 8th degree polynomial (what's inside the parentheses is a quadratic).  An 8th degree polynomial or lower is determined (or over-determined) by 9 points on it. We have 24 points shown on the polynomials--more than enough to show (prove) they're the same.

Posted by Charlie on 2017-12-10 10:32:30