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f(n)=sum(r=1)^(n) [r^(2)(""^(n)C(r)-""^(...

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

A

`f(30)=960`

B

`f(21)=483`

C

`f(16)=64`

D

`f(11)=44`

Text Solution

Verified by Experts

The correct Answer is:
A, B
`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.`
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