JEE Advance - Mathematics (1984 - No. 21)
If $$p$$ be a natural number then prove that $${p^{n + 1}} + {\left( {p + 1} \right)^{2n - 1}}$$ is divisible by $${p^2} + p + 1$$ for every positive integer $$n$$.
The statement is true for n=1.
Using the binomial theorem to expand (p+1)^(2n-1) is a valid approach.
Mathematical induction can be used to prove the statement.
The expression is not always divisible by p^2 + p + 1.
The statement is false for p = 1.