The sum of all positive integers n for which 1³+2³+ . .+(2 n)³ 1²+2²+ . .+n² is also an integer is.

Step-by-step solution

aligned 1³+2³+ . .+(2 n)³ 1²+2²+ . .+n² &= ( 2 n(2 n+1) 2 )² 6 n(n+1)(2 n+1) &= 6 n(2 n+1) n+1 &= 12 n²+6 n n+1 = 12 (n²-1 )+6(n+1)+6 n+1 &=1+ 6 n+1 aligned If the given terms is an intiger, then 6 n+1 must be an integer n =1,2,5 Sum =8