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Find the sum .^(n)C(1) + 2 xx .^(n)C(2)...

Find the sum `.^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "……" + n xx .^(n)C_(n)`.

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`S = 0 xx .^(n)C_(0) + 1 xx .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "……"`
`+ (n-1) xx .^(n)C_(1) + n xx .^(n)C_(n) "……"(1)`
`:. S = n xx .^(n)C_(n) + (n-1) xx .^(n)C_(n-1) + "……" + 1 xx .^(n)C_(1) + 0 xx .^(n)C_(0)`
or `S = n xx .^(n)C_(0) + (n-1) xx .^(n)C_(1) + "......"`
`+ 1 xx .^(n)C_(n-1) + 0 xx .^(n)C_(2) "....."(2)`
Adding (1) and (2), we get
`2S = n(.^(n)C_(0) + .^(n)C_(1) + .^(n)C_(2) + "......." + .^(n)C_(n)) = n xx 2^(n)`
`:. S = n xx 2^(n-1)`
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