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Find the sum sum(r=0)^(n) ""^(n+r)C(r )...

Find the sum `sum_(r=0)^(n) ""^(n+r)C_(r )`

Text Solution

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`underset(r=0)overset(n)sum.^(n+r)C_(r)=.^(n)C_(0)+.^(n+1)C_(1)+.^(n+2)C_(2)+"....."+.^(n+n)C_(n)`
`= .^(n)C_(n)+.^(n+1)C_(n)+"....."+.^(2n)C_(n)`
Coefficient of `x^(n)` in `{(1+x)^(n)+(1+x)^(n+1)+"...."+(1+x)^(2n)}`
`=` Coefficient of `x^(n)` in `((1+x)^n{1-(1+x)^(n+1)})/(1-(1+x))`
`=` Coefficient of `x^(n+1)` in `{(1+x)^(2n+1)-(1+x)^(n)}`
`= .^(2n+1)C_(n+1)`
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