`4C_0+4^2/2.c_1+4^3/3c_2+...............+(4^(n+1))/(n+1)C_n=(5^(n+1)-1)/(n+1)`

2.C_(0)+(2^(2)*C_(1))/(2)+(2^(3)*C_(2))/(3)+(2^(4)*C_(3))/(4)+.........+(2^(n+1)*C_(n))/(n+1)=(3^(n+1)-1)/(n+1)

2.C_(0)+(2^(2).C_(1))/(2)+(2^(3).C_(2))/(3)+(2^(4).C_(3))/(4)+......+(2^(n+1).C_(n))/(n+1)=(3^(n+1)-1)/(n+1)

If (1+x)^n=C_0+C_1x+C_2x^2+C_3x^3+...+C_nx^n then prove that 2.C_0+2^2C_1/2+2^3C_2/3+2^4C_3/4+...+2^(n+1)C_n/(n+1)=(3^(n+1)-1)/(n+1)

Statement-1: (C_(0))/(2.3)- (C_(1))/(3.4) +(C_(2))/(4.5)-.............+............+(-1)^(n) (C_(n))/((n+2)(n+3))= (1)/((n+1)(n+2)) Statement-2: (C_(0))/(k)- (C_(1))/(k+1) +(C_(2))/(k+3)+............+(-1)^(n) (C_(n))/(k+n)=int_(0)^(1)x^(k-1) (1 - x)^(n) dx

Statement-1: (C_(0))/(2.3)- (C_(1))/(3.4) +(C_(2))/(4.5)-.............+............+(-1)^(n) (C_(n))/((n+2)(n+3))= (1)/((n+1)(n+2)) Statement-2: (C_(0))/(k)- (C_(1))/(k+1) +(C_(2))/(k+3)+............+(-1)^(n) (C_(n))/(k+n)=int_(0)^(1)x^(k-1) (1 - x)^(n) dx

Prove that (C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n) = (C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)

(C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n)= (C_0C_1C_2.....C_(n-1) (n+1)^n)/(n!)

Prove that (C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n) = (C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)

Prove that (C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n) = (C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + … + C_(n) x^(n) , Show that (2^(2))/(1*2) C_(0) + (2^(3))/(2*3) C_(1) + (2^(4))/(3*4)C_(2) + ...+ (2^(n+2)C_n)/((n+1)(n+2))= (3^(n+2))/((n+1)(n+2))