9. `C_{0}-2^2C_{1}+3^2C_{2}-4^2C_{3}+.....+(-1)^n (n+1)^2C_{n} = 0`

Prove that C_(0)-2^(2)C_(1)+3^(2)C_(2)-4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)xx C_(n)=0 where C_(r)=^(n)C_(r)

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+C_(3)x^(2)+C_(4)x^(4)...+C_(n)x^(n),n>=0 prove that C_(0)-2^(2)C_(1)+3^(2)C_(2)+...+(-1)^(n)(n+1)^(2)C_(n)=0

If C_(0), C_(1), C_(2),..., C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then . 1^(2). C_(1) - 2^(2) . C_(2)+ 3^(2). C_(3) -4^(2)C_(4) + ...+ (-1).""^(n-2)n^(2)C_(n)= .

If C_(0) , C_(1) , C_(2) ,…, C_(n) are coefficients in the binomial expansion of (1 + x)^(n) and n is even , then C_(0)^(2)-C_(1)^(2)+C_(2)^(2)+C_(3)^(2)+...+ (-1)^(n)C_(n)""^(2) is equal to .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) + C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+...+C_(n)x^(n) then for n odd ,C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+...+(-1)C_(n)^(2), is equal to