If `(1+x)^n=C_0+C_1x+C2x2++C_n x^n , n in N ,t h e nC_0-C_1+C_2-+(-1)^(n-1)C_(m-1),` is equal to `(mltn)`

If (1+x)^n=c_0+C_1x+C_2x^2++C_n x^n , using derivtives prove t h a t C_1-2C_2+3C_3++(-1)^(n-1)nC_n=0

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_0 - C_2 + C_4 - C_6 + …….

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_1 - C_3 + C_5 + ……

If (1 + x)^n = C_0+C_1x+C_2x^2 + ... + C_nx^n ,then for n odd, C_0^2-C_1^2+C_2^2-C_3^2 + ... +(-1) C_n^2 , is equal to

If (1+x)^n=C_0+C_1x+C_2x^2+……..+C_nx^n , show that C_1/2+C_3/4+C_5/6+………..= (2^n-1)/(n+1)

If (1+x)^n=C_0+C_1x+C_2x^2+...+C_n x^n , t h e n C_0-(C_0+C_1)+(C_0+C_1+C_2)-(C_0+C_1+C_2+C_3)+ ...+(-1)^(n-1)(C_0+C_1+ C_(n-1)) , where n a) is even integer b) is a positive value c) a negative value d) divisible by 2^(n-1) divisible by 2^n

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 .

If (1+x)^n=C_0+C_1x+C_2x^2+...+C_nx^n then the value of (C_0)^2+(C_1)^2/2+(C_2)^2/3+...+(C_n)^2/(n+1) is equal to

If (1+x)^n=C_0+C_1x+C_2x^2+……..+C_nx^n in N prove that (a) 3 C_0- 8C_1+13C_2-18C_3+...."upto" (n+1) term=0 if n ge 2 (b ) 2C_0+2^2(C_1)/(2)+2^3(C_2)/(3)+2^4C_(3)/4+....+2^(n+1)(C_n)/(n+1)=(3^n+1-1)/(n+1) ( c) C_0^2+(C_1^2)/2+C_2^2/3+....+C_n^2/(n+1)=((2n+1)!)/(((n+1)!)^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 .