Prove that `(.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1)`.

Prove that .^(n)C_(1) + 2 .^(n)C_(2) + 3 .^(n)C_(3) + "…." + n . ^(n)C_(n) = n 2^(n-1) .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

Find the sum 1.^(n)C_(0) + 3 .^(n)C_(1) + 5.^(n)C_(2) + "….." + (2n+1).^(n)C_(n) .

Prove that .^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r ) .

If S_(n) = (.^(n)C_(0))^(2) + (.^(n)C_(1))^(2) + (.^(n)C_(n))^(n) , then maximum value of [(S_(n+1))/(S_(n))] is "_____" . (where [*] denotes the greatest integer function)

Prove that (.^(n)C_(1)sin2x+.^(n)C_(2)sin4x+.^(n)C_(3)sin6x+"…..")/(1+.^(n)C_(1)cos2x+.^(n)C_(2)cos4x+.^(n)C_(3)cos 6x+"……")

Prove that C_(0)^(2) + C_(1)^(2) + C_(2)^(2) +…+C_(n)^(2) = (2n!)/(n!)^(2)

The value of 2xx.^(n)C_(1) + 2^(3) xx .^(n)C_(3) + 2^(5) + "…." is