NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 4 - Principle of Mathematical Induction [Latest edition]

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

1 + 3 + 5 + ... + (2n – 1) = n² 

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

`sum_(t = 1)^(n - 1) t(t + 1) = (n(n - 1)(n + 1))/3`, for all natural numbers n ≥ 2.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

`(1 - 1/2^2).(1 - 1/3^2)...(1 - 1/n^2) = (n + 1)/(2n)`, for all natural numbers, n ≥ 2. 

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2²ⁿ – 1 is divisible by 3.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2n + 1 < 2ⁿ, for all natual numbers n ≥ 3.

Define the sequence a₁, a₂, a₃ ... as follows:
a₁ = 2, a_n = 5 a_n–1, for all natural numbers n ≥ 2.

Write the first four terms of the sequence.

Define the sequence a₁, a₂, a₃ ... as follows:
a₁ = 2, a_n = 5 a_n–1, for all natural numbers n ≥ 2.

Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula a_n = 2.5^n–1 for all natural numbers.

The distributive law from algebra says that for all real numbers c, a₁ and a₂, we have c(a₁ + a₂) = ca₁ + ca₂.

Use this law and mathematical induction to prove that, for all natural numbers, n ≥ 2, if c, a₁, a₂, ..., a_n are any real numbers, then c(a₁ + a₂ + ... + a_n) = ca₁ + ca₂ + ... + ca_n.

Prove by induction that for all natural number n sinα + sin(α + β) + sin(α + 2β)+ ... + sin(α + (n – 1)β) = `(sin (alpha + (n - 1)/2 beta)sin((nbeta)/2))/(sin(beta/2))`

Prove by the Principle of Mathematical Induction that 1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

Show by the Principle of Mathematical Induction that the sum S_n of the n term of the series 1² + 2 × 2² + 3² + 2 × 4² + 5² + 2 × 6² ... is given by

S_n = `{{:((n(n + 1)^2)/2",",  "if n is even"),((n^2(n + 1))/2",",  "if n is odd"):}`

Let P(n): “2ⁿ < (1 × 2 × 3 × ... × n)”. Then the smallest positive integer for which P(n) is true is ______.

A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also that P(5) is true. On the basis of this he could conclude that P(n) is true ______.

If P(n) : “2.4²ⁿ⁺¹ + 3³ⁿ⁺¹ is divisible by λ for all n ∈ N” is true, then the value of λ is ______.

If P(n): “49ⁿ + 16ⁿ + k is divisible by 64 for n ∈ N” is true, then the least negative integral value of k is ______.

State whether the following proof (by mathematical induction) is true or false for the statement.

P(n): 1² + 2² + ... + n² = `(n(n + 1) (2n + 1))/6`

Proof By the Principle of Mathematical induction, P(n) is true for n = 1,

1² = 1 = `(1(1 + 1)(2*1 + 1))/6`. Again for some k ≥ 1, k² = `(k(k + 1)(2k + 1))/6`. Now we prove that

(k + 1)² = `((k + 1)((k + 1) + 1)(2(k + 1) + 1))/6`

Give an example of a statement P(n) which is true for all n ≥ 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. 

Prove the statement by using the Principle of Mathematical Induction:

4ⁿ – 1 is divisible by 3, for each natural number n.

Prove the statement by using the Principle of Mathematical Induction:

2³ⁿ – 1 is divisible by 7, for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

n³ – 7n + 3 is divisible by 3, for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

3²ⁿ – 1 is divisible by 8, for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

For any natural number n, 7ⁿ – 2ⁿ is divisible by 5.

Prove the statement by using the Principle of Mathematical Induction:

For any natural number n, xⁿ – yⁿ is divisible by x – y, where x and y are any integers with x ≠ y.

Prove the statement by using the Principle of Mathematical Induction:

n³ – n is divisible by 6, for each natural number n ≥ 2.

Prove the statement by using the Principle of Mathematical Induction:

n(n² + 5) is divisible by 6, for each natural number n.

Prove the statement by using the Principle of Mathematical Induction:

n² < 2ⁿ for all natural numbers n ≥ 5.

Prove the statement by using the Principle of Mathematical Induction:

2n < (n + 2)! for all natural number n.

Prove the statement by using the Principle of Mathematical Induction:

`sqrt(n) < 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n)`, for all natural numbers n ≥ 2.

Prove the statement by using the Principle of Mathematical Induction:

2 + 4 + 6 + ... + 2n = n² + n for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

1 + 2 + 2² + ... + 2ⁿ = 2ⁿ⁺¹ – 1 for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

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

A sequence a₁, a₂, a₃ ... is defined by letting a₁ = 3 and a_k = 7a_{k – 1} for all natural numbers k ≥ 2. Show that a_n = 3.7^n–1 for all natural numbers.

A sequence b₀, b₁, b₂ ... is defined by letting b₀ = 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 d₁, d₂, d₃ ... is defined by letting d₁ = 2 and d_k = `(d_(k - 1))/"k"` for all natural numbers, k ≥ 2. Show that d_n = `2/(n!)` for all n ∈ N.

Prove that for all n ∈ N.
cos α + cos(α + β) + cos(α + 2β) + ... + cos(α + (n – 1)β) = `(cos(alpha + ((n - 1)/2)beta)sin((nbeta)/2))/(sin  beta/2)`.

Prove that, cosθ cos2θ cos2²θ ... cos2^n–1θ = `(sin 2^n theta)/(2^n sin theta)`, for all n ∈ N.

Prove that, sinθ + sin2θ + sin3θ + ... + sinnθ = `((sin ntheta)/2 sin  ((n + 1))/2 theta)/(sin  theta/2)`, for all n ∈ N.

Show that `n^5/5 + n^3/3 + (7n)/15` is a natural number for all n ∈ N.

Prove that `1/(n + 1) + 1/(n + 2) + ... + 1/(2n) > 13/24`, for all natural numbers n > 1.

Prove that number of subsets of a set containing n distinct elements is 2ⁿ, for all n ∈ N.

If 10ⁿ + 3.4ⁿ⁺² + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is ______.

For all n ∈ N, 3.5²ⁿ⁺¹ + 2³ⁿ⁺¹ is divisible by ______.

If xⁿ – 1 is divisible by x – k, then the least positive integral value of k is ______.

If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.

State whether the following statement is true or false. Justify.

Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.