$$A = \left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]$$ then by the principle of mathematical induction, prove that $${A^n} = \left[ {\matrix{ 1 & {2n} \cr 0 & 1 \cr } } \right]$$

Prove by induction that for all n $$\in$$ N, n² + n is an even integer (n $$\ge$$ 1).

Product of any r consecutive natural numbers is always divisible by

For each n $$\in$$ N, 2³ⁿ $$-$$ 1 is divisible by

here N is a set of natural numbers.

The number (101)¹⁰⁰ $$-$$ 1 is divisible by

The expression $2^{4 n}-15 n-1$, where $n \in \mathbb{N}$ (the set of natural numbers) is divisible by

Let $$P(n) = {3^{2n + 1}} + {2^{n + 2}}$$ where $$n \in N$$. Then