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For all positive integers n , show that ...

For all positive integers `n ,` show that `\ ^(2n)C_n+\ ^(2n)C_(n-1)=1/2(\ ^(2n+2)C_(n+1))` .

Text Solution

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`L.H.S = .^(2n)C_(n) + .^(2n)C_(n-1)`
`= .^(2n+1)C_(n) ( :' .^(n)C_(r) +.^(n)C_(r-1) = .^(n+1)C_(r))`
`= ((2n+1)!)/(n!(2n+1-n)!)`
`= ((2n+2)(2n+1)!)/((2n+2).n!(n+1)!)`
`= ((2n+1)!)/(2.(n+1).n!(n+1)!)`
`=(1)/(2).((2n+2)!)/((n+1)!(n+1)!)`
`= (1)/(2) .^(2n+1)C_(N+1) = R.H.S`. Hence Proved.
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