1^2+ (1^2+2^2 )+ (1^2+2^2+3^2 )+ + (1^2+2^2+ +n^2 )=

Step-by-step solution

aligned & 1^2+ (1^2+2^2 )+ (1^2+2^2+3^2 )+ & + (1^2+2^2+ +n^2 ) & = _ i=1 ^n (1^2+2^2+3^2+ +i^2 )= _ i=1 ^n _ j=1 ^i j^2 & = _ i=1 ^n i(i+1)(2 i+1) 6 n^2= n(n+1)(2 n+1) 6 & = _ i=1 ^n 2 i^3+3 i^2+i 6 = 1 3 _ i=1 ^n i^3+ 1 2 _ i=1 ^n i^2+ 1 6 _ i=1 ^n i & = 1 3 [ n(n+1) 2 ]^2+ 1 2 [ n(n+1)(2 n+1) 6 ]+ 1 6 [ n(n+1) 2 ] & array c n^3= [ n(n+1) 2 ]^2 ; n= n(n+1) 2 array & = 1 3 n^2(n+1)^2 4 + 1 2 n(n+1)(2 n+1) 6 + 1 6 n(n+1) 2 & = n^2(n+1)^2 12 + n(n+1)(2 n+1) 12 + n(n+1) 12 & = n^2(n+1)^2+n(n+1)(2 n+1)+n(n+1) 12 & = n