The value of ( _ n ( _ k=1 ^n k^3+6 k^2+11 k+5 (k+3)! ) ) is:

Step-by-step solution

aligned & _ n _ k=1 ^n k^3+6 k^2+11 k+5 (k+3)! & = _ n _ k=1 ^n k^3+6 k^2+11 k+6-1 (k+3)! & = _ n _ k=1 ^n (k+1)(k+2)(k+3)-1 (k+3)! & = _ n _ k=1 ^n (k+1)(k+2)(k+3) (k+3)! - 1 (k+3)! & = _ k=1 _ k=1 ^n ( 1 k! - 1 (k+3)! ) aligned aligned & = _ k=1 ( 1 1! + 1 2! + 1 3! + 1 4! + 1 n! - 1 4! - 1 5! - 1 6! .- 1 (n+3)! ) & = 1 1 + 1 2 + 1 6 = 10 6 = 5 3 aligned