Show that `C_0^2-C_1^2+C_2^2-C_3^2+...........+(-1)^n C_n^2=0` or `(-1)^(n/2)C_(n/2)` according as `n` is odd or even.

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) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

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

C0-(C1)/(2)+(C2)/(3)-............+(-1)^(n)(Cn)/(n+1)=(1)/(n+1)

C_(0)-(C_(1))/(2)+(C_(2))/(3)-............(-1)^(n)(C_(n))/(n+1)=(1)/(n+1)

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)

Prove that (^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

C_ (0) ^ (2) + 2C_ (1) ^ (2) + 3.C_ (2) ^ (2) + ............ + (n + 1) C_ (n ) ^ (2) =