The sum of all positive integers n for which 13+23....+(2n)312+22+...n2 is also an integer is.
8
13+23....+(2n)312+22+...n2=(2n(2n+1)22)26n(n+1)(2n+1)
=6n(2n+1)n+1
=12n2+6nn+1=12(n2+1)+6(n+1)+6n+1
=12n−6+6n+1
If the given terms are an integers, then 6n+1 must be an integer
⇒n=1,2,5
Sum=8