Prove that P(n,n) = 2P(n,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 : (2n) ! = 2^n (n!)[1.3.5.... (2n-1)] for all natural numbers n.

Prove that : "^nP_n= 2^nP_(n-2) .

Let P(n) be the statement " n^2-n+41 " is prime. Prove that P(1), P(2) and P(3) are true. Also prove that P(41) is not true. How does this not contradict the Principle of Induction ?

If a,b,c,d are in G.P., prove that (a^(n) + b^(n)), (b^(n) + c^(n)), (c^(n) + d^(n)) are in G.P.

Prove that : ^(2n)C_n = (2^n [1.3.5. ..........(2n-1)])/(n!) .

Prove that : 7^(2n)+(2^(3n-3))(3^(n-1)) is divisible by 25 forall n in N .

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

If P (n) is the statement : "n (n + 1) (n + 2) is divisible by 12”, prove that P(3) and P(4) are true, but P(5) is not true.

Let P(n) denote the statement that n^2+n is odd . It is seen that P(n)rArr P(n+1),P(n) is true for all