Principle Of Mathematical Induction CBSE Questions & Answers

The inequality \({2^n} < n!,n \in N\) is true for :

For each \(n \in N,{a^{2n - 1}} + {b^{2n - 1}}\) is divisible by :

For each n \( \in \) N , n (n + 1 ) ( 2n + 1 ) is divisible by :

For each n \( \in \) N , \(3({5^{2n + 1}}) + {2^{3n + 1}}\)is divisible by :

For each n \( \in \) N , \({2^{3n}} - 1\)is divisible by :

For each n \( \in \) N , \({3^{2n}} - 1\) is divisible by :

The statement P ( n ) : “\({\left( {n + 3} \right)^2} > {2^{n + 3}}\) “ is true for :

The smallest positive integer ‘ n ‘ for which \({2^n}\left( {1 \times 2 \times 3 \times ............... \times n} \right) < {n^n}\) holds is :

If \({10^n} + 3 \times {4^{n + 1}} + k\) is divisible by 9 for all n \( \in \) N , then the least positive integral value of k is :

The smallest positive integer n , for which \(\left( {{\text{ 1 }} \times {\text{ 2 }} \times {\text{ 3 }} \times \ldots \ldots \times {\text{n }}} \right){\text{ }} < \)\({\left( {{{n + 1} \over 2}} \right)^n}\) holds :

Let \(P\left( n \right):{n^2} - n + 41\) is a prime number . then :

Let that \(P\left( n \right) \Rightarrow P\left( {n + 1} \right)\) for all natural numbers n. also , if P ( m ) is true , m \(\in \) N , then we conclude that

Consider the statement P ( n ) : “\({n^2} \ge 100\) “ . Here \(P\left( n \right) \Rightarrow P\left( {n + 1} \right)\) for all natural numbers n . Does it mean

The smallest positive integer ‘ n ‘ for which P ( n ) : \({2^n} < \left( {1 \times 2 \times 3 \times ............ \times n} \right)\) holds is :

If \({x^{n}} - 1\) is divisible by x – k for all n belongs to natural numbers N , then the least positive integral value of k is :