`f(n)=sum_(r=1)^(n) [r^(2)(""^(n)C_(r)-""^(n)C_(r-1))+(2r+1)(""^(n)C_(r ))]`, then

Options

A. `f(30)=960`
B. `f(21)=483`
C. `f(16)=64`
D. `f(11)=44`

Answer

A, B

Explanation

`f(n)=underset(r=1)overset(n)sum[r^(2)(.^nC_(r)-.^nC_(r=1))+(2r).^nC_(r)+.^nC_(r)]`
`=underset(r-1)overset(n)sum[(r^(2)+2r+1).^nC_(r)-r^(2).^nC_(r-1)]`
`=underset(r=1)overset(n)sum[(r+1)^(2).^nC_(r)-r^(2).^nC_(r=1)]=underset(r=1)overset(n)sum[V_(r-1)-V_(r)]`
`=V_(2)-V_(1)+V_(3)-V_(2)+...+V_(n+1)-V_(n)`
`=V_(n+1)-V_(1)=(n+1)^(2).^(n)C_(n)-1=n^(2)+2n.`