Question

If p be a natural number, then prove that p^{n + 1} + (p + 1)^{2n - 1} is divisible by p² + p + 1 for every positive integer n.

Answer 1

Let P(n) : P^{n + 1} + (p + 1)^{2n - 1} is divisible by p² + p + 1.

For n = 1, P(1): p² +(p+1)¹

which is divisible by p² + p + 1

.'. P (1)is true.

Let P (k) be true. ie,

.'. P (k+ 1) is divisibleby p² + p + 1

.'. P (k + 1) is true.

Hence, by mathematical induction P (n) is true for all n ∈ N.