`(C_(0))/(1.3)-(C_(1))/(2.3)+(C_(2))/(3.3)-(C_(3))/(4.3)+.......+(-1)^(n)(C_(n))/((n+1)*3)`

If C_(0), C_(1) C_(2) ….., denote the binomial coefficients in the expansion of (1 + x)^(n) , then (C_(0))/(2) - (C_(1))/(3) + (C_(2))/(4)- (C_(3))/(5)+...+ (-1)^(n)(C_(n))/(n+2) =

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

Show that (C_(0))/(1) - (C_(1))/(4) + (C_(2))/(7) - … + (-1)^(n) (C_(n))/(3n +1) = (3^(n) * n!)/(1*4*7…(3n+1)) , where C_(r) stands for ""^(n)C_(r) .

If C_(0), C_(1), C_(2),..., C_(n) are binomial coefficients in the expansion of (1 + x)^(n), then the value of C_(0) + (C_(1))/(2) + (C_(2))/(3) + (C_(3))/(4) +...+ (C_(n))/(n+1) is

Find .^(n)C_(1)-(1)/(2).^(n)C_(2)+(1)/(3).^(n)C_(3)- . . . +(-1)^(n-1)(1)/(n).^(n)C_(n)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + …+ C_(n) x^(n) , show that C_(1) - (C_(2))/(2) + (C_(3))/(3) - …(-1)^(n-1) (C_(n))/(n) = 1 + (1)/(2) + (1)/(3) + …+ (1)/(n) .

If (1+x)^(n)=C_(0)+C_(1)x+…..+C_(n)x^(n) , then (C_(1))/(C_(0))+(2C_(2))/(C_(1))+(3C_(3))/(C_(2))+....+(nC_(n))/(C_(n-1)) is :

if (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a++.^(n)C_(2)a^(2)+ . . .+.^(n)C_(n)a^(n) , then prove that (.^(n)C_(1))/(.^(n)C_(0))+(2(.^(n)C_(2)))/(.^(n)C_(1))+(3(.^(n)C_(3)))/(.^(n)C_(2))+. . . +(n(.^(n)C_(n)))/(.^(n)C_(n-1))= Sum of first n natural numbers.

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

Prove that 1/(n+1)=(.^n C_1)/2-(2(.^n C_2))/3+(3(.^n C_3))/4- . . . +(-1^(n+1))(n*(.^n C_n))/(n+1) .