_ n [ array l 1 n + n^2 (n+1)^3 + n^2 (n+2)^3 + n^2 (n+3)^3 + + 1 125 n array ]=

Step-by-step solution

Given, aligned & _ n [ 1 n + n^2 (n+1)^3 + n^2 (n+2)^3 + n^2 (n+3)^3 + 1 125 n ] & = _ n [ array l n^2 (n+0)^3 + n^2 (n+1)^3 + n^2 (n+2)^3 + n^2 (n+3)^3 + + n^2 (n+4 n)^3 array ] & = _ n _ r=0 ^ 4 n n^2 (n+r)^3 aligned aligned & Let r n =x and 1 n =d x= ₀^4 d x (1+x)^3 = [ (1+x)⁻² -2 ]₀^4 & =- 1 2 [ 1 5^2 -1 ]= 1 2 (1- 1 25 )= 12 25 . aligned Hence, option (c) is correct.