Prove that
`1/(m!).^(n)C_(0)+(n)/((m+1)!).^(n)C_(1)+(n(n-1))/((m+2)!).^(n)C_(2)+"....."+(n(n-1)"...."2xx1)/((m+2)!).^(n)C_(n) = ((m+n+1)(m+n+2)"....."(m+2n))/((m+n)!)`

Let m, in N and C_(r) = ""^(n)C_(r) , for 0 le r len Statement-1: (1)/(m!)C_(0) + (n)/((m +1)!) C_(1) + (n(n-1))/((m +2)!) C_(2) +… + (n(n-1)(n-2)….2.1)/((m+n)!) C_(n) = ((m + n + 1 )(m+n +2)…(m +2n))/((m +n)!) Statement-2: For r le 0 ""^(m)C_(r)""^(n)C_(0)+""^(m)C_(r-1)""^(n)C_(1) + ""^(m)C_(r-2) ""^(n)C_(2) +...+ ""^(m)C_(0)""^(n)C_(r) = ""^(m+n)C_(r) .

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 ) .

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

Prove that (1^(2))/(3).^(n)C_(1)+(1^(2) + 2^(2))/(7).^(n)C_(2)+(1^(2)+2^(2)+3^(2))/(7).^(n)C_(3)+"...." +(1^(2)+2^(3)+"....."+n^(2))/(2n+1).^(n)C_(n) = (n(n+3))/(6)2^(n-2) .

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

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

Prove that .^(2n)C_(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)

Prove that tan^(-1)((m)/(n))-tan^(-1)((m-n)/(m+n))=(pi)/(4).

Prove that ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot