Prove that `C_(1)^(2) - 2. C_(2)^(2) + 3. C_(3)^(2) - …. -2n . C_(2n)^(2) = (-1)^(n). C_(n)`

Explanation

We know that , `(1 + x )^(2n) = C_(0) + C_(1)x + C_(2) x^(2) + …+ C_(2n) x^(2n)`
On differntiating both sides w.r.t.x, we get
` 2n (1 + x)^(2n-1) = C_(1) + 2.C_(2) x + 3. C_(3) x^(3) + …+ 2nC_(2n x^(2n-1) ` …(i)
and ` (1 - (1)/(x))^(2n) = C_(0) - C_(1).(1)/(x) + C_(2) .(1)/(x^(2)) C_(3) . (1)/(x^(3)) + ...+ C_(2n). (1)/(x^(2n))` ...(ii)
On multiplying Eqs. (i) and (ii) , we get
` 2n (1 + x)^(2n-1) (1 - (1)/(x))^(2n)`
= `[C_(1) + 2*C_(2)x + 3. C_(3) x^(2) +...+ 2n*C_(2n)x^(2n-1)] xx[C_(0) - C_(1) ((1)/(x)) + C_(2) ((1)/(x^(3))) -...+C_(2n) ((1)/(x^(2n)))]`
Coefficent of ` ((1)/(x))` on the LHS
= Coefficient of ` (1)/(x) `in 2n `((1)/(x^(2n)))(1 + x)^(2n - 1) (x - 1)^(2n)`
Coefficient if ` x^(2n - 1)` in 2n ` (1 - x^(2))^(2-1) (1 - x)`
` 2n (-1)^(n-1) . (2n -1) C_(n-1) (-1)`
` = (-1)^(n) (2n) ((2n -1)!)/((n-1)!n!) = (-1)^(n)n((2n)!)/((n!^(2)))n`
` = - (-1)^(n)n . C_(n)` ...(iii)
Again , the coefficient of `((1)/(x))` on the RHS
` = - (C_(1)^(2) - 2 *C_(2)^(2) + 3*C_(3)^(2)-...-2nC_(2n)^(2))` ...(iv)
From Eqs. (iii) and (iv),
`- C_(1)^(2) - 2 *C_(2)^(2) + 3*C_(3)^(2)-...-2nC_(2n)^(2) = (-1)^(n) n.C_(n)`.