C_(0)^(2)+3.C_(1)^(2)+5.C_(2)^(2)+..........+(2n+1)*C_(n)^(2)=

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)^(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 .

prove that :C_(0)^(2)+3C_(1)^(@)+5C_(2)^(2)+...+(2n+1)C_(n)^(2)=((n+1)2n!)/((n!)^(2))

(*^(n)c_(0))^(2)+3(.^(n)C_(1))^(2)+5(.^(n)C_(2))^(2)+ ......+(2n+1)(.^(n)C_(n))^(2)

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

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : C_(0)^(2)+(C_(1)^(2))/(2)+(C_(2)^(2))/(3)+.....+(C_(n)^(2))/(n+1)=((2n+1)!)/({(n+1)!}^(2))