Prove that P(n,r) = C(n,r) P(r,r)

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^(2)R^(n) = S^(n) .

Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))

Prove that : ^nP_r= "^(n-1)P_r+r ^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

If P(n) is the statement “sum of first n natural numbers is divisible by n + 1", prove that P(r + 1) is true if P(r) is true.

Verify that ""^nC_r= frac (n)(r) "^(n-1)C_(r-1) and hence prove that "^nC_r= (n!)/(r!(n-r)!) .

Prove that : ^nP_r= n"^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

Prove that: sum_(r=0)^(n) 3^( r) ""^(n)C_(r) = 4^(n) .

Prove that "^nC_r+2 ^(n)C_(r-1)+ ^(n)C_(r-2) = ^(n+2)C_r .

Prove that (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!)= ((n+1)!)/ (r!(n-r+1)!) .