Prove that P(n,r) = nP(n-1,r-1)

Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(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 : "^nP_n= 2^nP_(n-2) .

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 : ^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 the coefficient of the rth, (r+1)th and (r+2)th terms in the expansion of (1+x)^(n) are in A.P., prove that n^(2) - n(4r +1) + 4r^(2) - 2=0 .

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)!) .

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.

If t_r is the rth term is the expansion of (1 + a)^n , in ascending powers of a, prove that : r(r+1) t_(r+2)= (n-r+1)(n-r) a^2 t_r .