Mathematical Induction - Algebra - Mathematics Previous Year Questions

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

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

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

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

here N is a set of natural numbers.

Product of any r consecutive natural numbers is always divisible by

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

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