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Find the sum sum(r=1)^n r^2(^n Cr)/(^n C...

Find the sum `sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1))` .

Text Solution

Verified by Experts

The correct Answer is:
`(n(n+1)(n+2))/(6)`
`t_(r)=r^(2)(.^(n)C_(r))/(.^(n)C_(r)-1)`
`=r^(2)(n-r+1)/(r)`
`=r(n+1-r)`
`=(n+1)r-r^(2)`
`:.` Sum `= (n+1)underset(r=1)overset(n)sumr-underset(r=1)overset(n)sumr^(2)`
`=(n+1)(n(n+1))/(2)-(n(n+1)(2n+1))/(6)`
`= (n(n+1))/(6){3(n+1) - (2n+1)}`
`= (n(n+1)(n+2))/(6)`
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