Find the sum `sum_(r=0)^n^(n+r)C_r` .

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .

Find the sum sum_(i=0)^r.^(n_1)C_(r-i) .^(n_2)C_i .

If sum_(r=1)^n T_r=(3^n-1), then find the sum of sum_(r=1)^n1/(T_r) .

Find the sum sum_(r =1)^(oo) tan^(-1) ((2(2r -1))/(4 + r^(2) (r^(2) -2r + 1)))

If sum_(r=1)^n T_r=n(2n^2+9n+13), then find the sum sum_(r=1)^nsqrt(T_r)dot

Find the sum Sigma_(r=1)^(n) r/((r+1)!) . Also, find the sum of infinite terms.

Find the sum Sigma_(r=1)^(n) 1/(r(r+1)(r+2)(r+3)) Also,find Sigma_(r=1)^(oo) 1/(r(r+1)(r+2)(r+3))

Consider a G.P. with first term (1+x)^(n) , |x| lt 1 , common ratio (1+x)/(2) and number of terms (n+1) . Let 'S' be sum of all the terms of the G.P. , then sum_(r=0)^(n)"^(n+r)C_(r )((1)/(2))^(r ) equals

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .