Principle Of Mathematical Induction CBSE Questions & Answers

Let \(P\left( n \right)\) be a statement and let \(P\left( n \right) \Rightarrow P\left( {n + 1} \right)\) for all natural numbers n , then \(P\left( n \right)\) is true for

Let \(P\left( n \right)\) b e a statement \({2^n} < n!\) , where n is a natural number , then \(P\left( n \right)\) is true for

If x \(>\) -1 , then the statement \({\left( {1 + x} \right)^n} > 1 + nx\) is true for

The smallest +ve integer n , for which\( n! < \)\({\left( {{{n + 1} \over 2}} \right)^n}\) holds is

If P ( n ) = 2+4+6+………………..+2n , n \( \in \) N , then P ( k ) = k ( k + 1 ) + 2 \( \Rightarrow \) P ( k + 1 ) = ( k + 1 ) ( k +2 ) + 2 for all k \( \in \) N . So we can conclude that P ( n ) = n ( n + 1 ) +2 for

\(x\left( {{x^{n - 1}} - n{a^{n - 1}}} \right) + {a^n}\left( {n - 1} \right)\) is divisible by \({\left( {x - a} \right)^2}\) for

The greatest positive integer , which divides n ( n + 1 ) ( n + 2 ) ( n + 3 ) for all n \( \in \) N , is

Let P ( n ) denote the statement \({n^2} + n\) is odd , It is seen that \(P\left( n \right) \Rightarrow P\left( {n + 1} \right)\) , therefore P ( n ) is true for all

The greatest positive integer, which divides \(\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)..................\left( {n + r} \right)\forall n \in W\) , is

The statement P ( n ) : “\({\text{1 X 1}}!{\text{ }} + {\text{ 2 X 2}}!{\text{ }} + {\text{ 3 X 3}}!{\text{ }} + {\text{ }} \ldots .. + {\text{ n X n}}!{\text{ }} = {\text{ }}\left( {{\text{ n }} + {\text{ 1 }}} \right){\text{ }}!--{\text{ 1}}\) “ is

A student was asked to prove a statement P ( n ) by method of induction . He proved that P ( 3 ) is true and that \(P\left( n \right) \Rightarrow P\left( {n + 1} \right)\) for all natural numbers n . On the basis of this he could conclude that P ( n ) is true

\({3^{2n + 2}} - 8n - 9\) is divisible by 64 for all

The inequality \(n! > {2^{n - 1}}\) is true

The statement \({3^n} > 4n\) is true for all

The statement \({2^n} > 3n\) is true for all