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.

Start with RHS = 2(n/2 (n + 1))^4 = 1/8 (n^2 + n)^4

Then LHS, considering each bracket separately:

Sum (1 to n) n^5 = 1/6 (n^2 + n)^3 - 1/12 (n^2 + n)^2

Sum (1 to n) n^7 = 1/8 (n^2 + n)^4 - 1/6 (n^2 + n)^3 + 1/12 (n^2 + n)^2

So the n^5 term exactly cancels out of the n^7 term, leaving 1/8 (n^2 + n)^4, as was to be shown. Very neat.

Posted by broll on 2017-12-10 21:33:38