If `(1+x)^n = C_0 + C_1x + ..... + C_n x^n ( n in N) ` prove that

If (1+x)^(n)=C_(0)+C_(1)x+.....+C_(n)x^(n)(n in N) prove that

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + 2C_1 + ….. + 2 ""^nC_n = 3^n

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + (C_1)/(2) + (C_2)/(3) + ……. + (C_n)/(n+1) = (2^(n+1) -1)/(n+1)

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove that : C_0+ 2C_1 +.........+2""^nC_n=3^n .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove that : C_0+ C_1/2 +C_2/3+.........+C_n/(n+1)= (2^(n+1)-1)/(n+1) .

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

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that (1*2) C_(2) + (2*3) C_(3) + …+ {(n-1)*n} C_(n) = n(n-1) 2^(n-2) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that (1*2) C_(2) + (2*3) C_(3) + …+ {(n-1)*n} C_(n) = n(n-1) 2^(n-2) .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0C_n+C_1C_(n-1)+C_2C_(n-2)+.....+ C_nC_0= ((2n!))/(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 .