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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_(1)^(2) + 2C_(2)^(2) + 3C_(3)^(2) + ..+ nC_(n)^(2) = ((2n-1)!)/(((n-1)!)^(2))`

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Verified by Experts

Given ` (1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2)x^(2) + C_(3)x^(3) + …+ C_(n) x^(n)`
Differentiating both sides w.r.t.x, we get
`n(1+ x)^(n-1) = 0+C_(1)+ 2C_(2) x + 3C_(3)x^(2) + ...+ nC_(n) x^(n-1) `
`rArr n(1+ x)^(n-1) = C_(1)+ 2C_(2) x + 3C_(3)x^(2) + ...+ nC_(n) x^(n-1) `...(i)
and `rArr n(1+ x)^(n) = C_(0)x^(n) + C_(1)^(n-1) + C_(2)x^(n-2) + C_(3) x^(n-3)+...+ C_(n) `...(ii)
On multiplying Eqs. (i) and(ii) , then we get
` n(1+ x)^(2n-1) =( C_(1)+ 2C_(2) x + 3C_(3) x^(2) + ...+ nC_(n) x^(n-1))`
` xx(C_(0) x^(n) + C_(1) x^(n-1) + C_(2) x^(n-2) + C_(3) x^(n-3) + ...+ C_(n))` ...(iii)
Now , coefficient of ` x^(n-1) ` on RHS
` = C_(1)^(2) + 2C_(2)^(2) + 3C_(3)^(2) + ...+ nC_(n)^(2)`
and coefficient of `x^(n-1)` on LHS
`= n*""^(2n-1)C_(n-1) = n*((2n-1)!)/((n-1)!n!)`
` = ((2n-1)!)/((n-1)!(n-1)!) = ((2n-1)!)/({(n-1)!}^(2))`
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