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If (1 + x)^(n) = C(0) + C(1) x + C(2)x^...

If ` (1 + x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + C_(3) x^(3)+ …+ C_(n) x^(n)` , prove that
` C_(0) - 3C_(1) + 5C_(2) - …+ (-1)^(n) (2n-1)C_(n) = 0 `

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Verified by Experts

`T_(r)=(-1)^r(2r+1)^nC_(r)=2(-1)^rr.""^(n)C_(r)+(-1)^r ""^(n)C_(r)`
`SigmaT_(r)=2underset(r=1)overset(n)Sigma(-1)^r.r.n/r.^(n-1)C_(r-1)+underset(r=0)overset(n)Sigma(-1)^r""^nC_r=2underset(r=0)overset(n)Sigma(-1)^r ""^(n-1)C_(r-1)+underset(r=1)overset(n)Sigma(-1)^r.""^(n)C_(r)`
`=2[""^(n-1)C_(0)""^(n-1)C_(1)+....]+[""^nC_(0)-""^nC_(1)+....]=0`
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