aligned & _ n n [ 1 (3 n^2+8 n+4 ) + 1 3 n^2+16 n+16 + . & .+ 1 15 n^2 ]= aligned

Step-by-step solution

aligned & _ n n [ 1 3 n^2+8 n+4 + 1 3 n^2+16 n+16 + + 1 15 n^2 ] & = _ n n _ r=1 ^n 1 3 n^2+8 n r+(2 r)^2 & = _ n 1 n _ r=1 ^n 1 3+8 ( r n )+4 ( r n )^2 & = ₀^1 d x 3+8 x+4 x^2 = ₀^1 d x 4(x+1)^2-1 & = 1 4 ₀^1 d x (x+1)^2- ( 1 2 )^2 = 1 4 1 2 1 2 [ (x+1)- 1 2 (x+1)+ 1 2 ]₀^1 & = 1 4 [ 3 / 2 5 / 2 - 1 / 2 3 / 2 ]= 1 4 9 5 . aligned