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Prove that (r+1)xx.^(n)C(0)-rxx.^(n)C(...

Prove that
`(r+1)xx.^(n)C_(0)-rxx.^(n)C_(1)+(r-1)xx.^(n)C_(2)-(r-2).^(n)C_(3) + "...."+(-1)^(r)xx.^(n)C_(r)+"...."=(-1)^(r)xx.^(n-2)C_(r)`.

Text Solution

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`(r+1)xx.^(n)C_(0)-rxx.^(n)C_(1)+(r-1)xx.^(n)C_(2)-(r-2).^(n)C_(3)+"……"+(-1)^(r ) xx .^(n)C_(r ) + "….."`
`=` Coefficient of `x^(r )` in
`(.^(n)C_(0)--.^(n)C_(1)x+.^(n)C_(2)x^(2)-.^(n)C_(3)x^(3)+"…….."(-1)^(r)xx.^(n)C_(r )+"….") xx(1+2x+3x^(2)+4x^(3)+"....."(r+1)x^(r)+".....")`
`=` Coefficient of `x^(r)` in `(1-x)^(n)(1-x)^(2)`
= Coefficientof `x^(r)` in `(1-x)^(n-2)`
`= (-1)^(r)xx.^(n-2)C_(r )`
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