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_nx^n prove the following : 2C_0+5 C_1+8C_2+...+(3n+2)C_n=(3n+4)2^(n-1) .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0-2C_1+3C_2-.........+(-1)^n (n+1)C_n=0 .

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

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : (1+C_1/C_0)(1+C_2/C_1)(1+C_3/C_2).... (1+C_n/ C_(n-1))=((n+1)^n)/(n!) .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0+1/3 C_2+1/5C_4+1/7C_6+.......... = 2^n/(n+1) .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : 2C_0+2^2 C_1/2+2^3 C_2/3+........+2^(n+1) C_n/ (n+1)= (3^(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_(1) + 2C_(2) + 3C_(3) + ...+ n""C_(n) = n*2^(n-1)

If (1+x)^n = C_0 +C_1x+C_2x^2+……… +C_nx^n , then prove that : C_1 -2C_2+………….+(-1)^(n-1)nC_n =0