NCERT Exemplar Solutions
Class 11 - Mathematics

Chapter 4 : PRINCIPLE OF MATHEMATICAL INDUCTION

Give an example of a statement \(P(n)\) which is true for all \(n \ge 4\) but \(P(1)\), \(P(2)\) and \(P(3)\) are not true. Justify your answer.

Give an example of a statement \(P(n)\) which is true for all \(n\). Justify your answer.

\(4^n - 1\) is divisible by 3, for each natural number \(n\).

\(2^{3n} - 1\) is divisible by 7, for all natural numbers \(n\).

\(n^3 - 7n + 3\) is divisible by 3, for all natural numbers \(n\).

\(3^{2n} - 1\) is divisible by 8, for all natural numbers \(n\).

For any natural number \(n\), \(7^n - 2^n\) is divisible by 5.

For any natural number \(n\), \(x^n - y^n\) is divisible by \(x - y\), where \(x\) and \(y\) are integers with \(x \ne y\).

\(n^3 - n\) is divisible by 6, for each natural number \(n \ge 2\).

\(n(n^2 + 5)\) is divisible by 6, for each natural number \(n\).

\(n^2 < 2^n\) for all natural numbers \(n \ge 5\).

\(2n < (n + 2)!\) for all natural numbers \(n\).

\(\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + \dots + \dfrac{1}{\sqrt{n}}\), for all natural numbers \(n \ge 2\).

\(2 + 4 + 6 + \dots + 2n = n^2 + n\) for all natural numbers \(n\).

\(1 + 2 + 2^2 + \dots + 2^n = 2^{n+1} - 1\) for all natural numbers \(n\).

\(1 + 5 + 9 + \dots + (4n - 3) = n(2n - 1)\) for all natural numbers \(n\).

A sequence \( a_1, a_2, a_3, \ldots \) is defined by letting \( a_1 = 3 \) and \( a_k = 7a_{k-1} \) for all natural numbers \( k \ge 2 \). Show that \( a_n = 3 \cdot 7^{n-1} \) for all natural numbers.

A sequence \( b_0, b_1, b_2, \ldots \) is defined by letting \( b_0 = 5 \) and \( b_k = 4 + b_{k-1} \) for all natural numbers \( k \). Show that \( b_n = 5 + 4n \) for all natural number \( n \) using mathematical induction.

A sequence \( d_1, d_2, d_3, \ldots \) is defined by letting \( d_1 = 2 \) and \( d_k = \dfrac{d_{k-1}}{k} \) for all natural numbers \( k \ge 2 \). Show that \( d_n = \dfrac{2}{n!} \) for all \( n \in \mathbb{N} \).

Prove that for all \( n \in \mathbb{N} \)

\(\cos \alpha + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \cdots + \cos(\alpha + (n-1)\beta) = \dfrac{\cos\left(\alpha + \dfrac{n-1}{2}\beta\right) \sin\left(\dfrac{n\beta}{2}\right)}{\sin\left(\dfrac{\beta}{2}\right)}\).

Prove that \(\cos \theta \cos 2\theta \cos 2^2\theta \cdots \cos 2^{n-1}\theta = \dfrac{\sin 2^n \theta}{2^n \sin \theta}\), for all \( n \in \mathbb{N} \).

Prove that \( \sin \theta + \sin 2\theta + \sin 3\theta + \cdots + \sin n\theta = \dfrac{\sin\left(\dfrac{n\theta}{2}\right) \sin\left(\dfrac{(n+1)\theta}{2}\right)}{\sin\left(\dfrac{\theta}{2}\right)} \), for all \( n \in \mathbb{N} \).

Show that \( \dfrac{n^5}{5} + \dfrac{n^3}{3} + \dfrac{7n}{15} \) is a natural number for all \( n \in \mathbb{N} \).

Prove that \( \dfrac{1}{n+1} + \dfrac{1}{n+2} + \cdots + \dfrac{1}{2n} > \dfrac{13}{24} \), for all natural numbers \( n > 1 \).

Prove that the number of subsets of a set containing \( n \) distinct elements is \( 2^n \), for all \( n \in \mathbb{N} \).

If \(P(n) : 2n < n!\), \(n \in \mathbb{N}\), then \(P(n)\) is true for all \(n \ge ____\).

Let \(P(n)\) be a statement and let \(P(k) \Rightarrow P(k+1)\), for some natural number \(k\), then \(P(n)\) is true for all \(n \in \mathbb{N}\).