If p is a natural number, then prove that `p^(n+1) + (p+1)^(2n-1)` is divisible by `p^(2) + p +1` for every positive integer n.

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

If p is a fixed positive integer, prove by induction that p^(n +1) + (p + 1)^(2n - 1) is divisible by P^2+ p +1 for all n in N .

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

Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

Prove that : P(n,n)= 2P (n,n -2)

Prove that ^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

Prove that .^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

If P(n) stands for the statement n(n+1) (n+2) is divisible by 6, then what is p(3)?

If P(n) be the statement " 10n+3 is a prime number", then prove that P(1) and P(2) are true but P(3) is false.