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sum(r=1)^n 1/((r+1)(r+2)) .^(n+3)Cr=...

`sum_(r=1)^n 1/((r+1)(r+2)) .^(n+3)C_r=`

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Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

sum_(r=1)^(n) {sum_(r1=0)^(r-1) ""^(n)C_(r) ""^(r)C_(r_(1)) 2^(r_1)} is equal to

sum_(r=1)^(n) {sum_(r1=0)^(r-1) ""^(n)C_(r) ""^(r)C_(r_(1)) 2^(r1)} is equal to

sum_(r=1)^(n) {sum_(r1=0)^(r-1) ""^(n)C_(r) ""^(r)C_(r_(1)) 2^(r1)} is equal to

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}