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

Given that: d1 = 2 and dk = `(d_(k - 1))/k` Let P(n): dn = `2/(n!)` Step 1: P(1) : d1 = `2/(1!)` = 2 which is true for P(1). Step 2: P(k) : dk = `2/(k!)`. Let it be true for P(k). Step 3: Given that: dk = `(d_(k - 1))/k` &there4; dk+1 = `(d_(k + 1 - 1))/(k + 1) = d_k/(k + 1)` ⇒ dk+1 = `1/(k + 1) . d_k = 1/(k + 1) . 2/(k!)` ⇒ dk+1 = `2/((k + 1)!)` Which is true for P(k + 1). Hence, P(k + 1) is true whenever P(k) is true.

A sequence d1, d2, d3 ... is defined by letting d1 = 2 and dk = kdk-1k for all natural numbers, k ≥ 2. Show that dn = 2n! for all n ∈ N.

Question

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.

Theorem

Solution

Given that: d₁ = 2 and d_k = `(d_(k - 1))/k`

Let P(n): d_n = `2/(n!)`

Step 1: P(1) : d₁ = `2/(1!)` = 2 which is true for P(1).

Step 2: P(k) : d_k = `2/(k!)`. Let it be true for P(k).

Step 3: Given that: d_k = `(d_(k - 1))/k`

∴ d_k+1 = `(d_(k + 1 - 1))/(k + 1) = d_k/(k + 1)`

⇒ d_k+1 = `1/(k + 1) . d_k = 1/(k + 1) . 2/(k!)`

⇒ d_k+1 = `2/((k + 1)!)` Which is true for P(k + 1).

Hence, P(k + 1) is true whenever P(k) is true.

shaalaa.com

Principle of Mathematical Induction

Is there an error in this question or solution?

Q 19 Q 18 Q 20

Chapter 4: Principle of Mathematical Induction - Exercise [Page 71]

APPEARS IN

NCERT Exemplar Mathematics Exemplar [English] Class 11

Chapter 4 Principle of Mathematical Induction
Exercise | Q 19 | Page 71

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Principle of Mathematical Induction

video tutorial00:17:42

RELATED QUESTIONS

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.2² + 3.2² + … + n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2

Prove the following by using the principle of mathematical induction for all n ∈ N:

(1+3/1)(1+ 5/4)(1+7/9)...`(1 + ((2n + 1))/n^2) = (n + 1)^2`

Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 3.

Prove the following by using the principle of mathematical induction for all n ∈ N: 10²ⁿ^{– 1}+ 1 is divisible by 11

If P (n) is the statement "n² + n is even", and if P (r) is true, then P (r + 1) is true.

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.

1² + 2² + 3² + ... + n² =\[\frac{n(n + 1)(2n + 1)}{6}\] .

1.3 + 3.5 + 5.7 + ... + (2n − 1) (2n + 1) =\[\frac{n(4 n^2 + 6n - 1)}{3}\]

n(n + 1) (n + 5) is a multiple of 3 for all n ∈ N.

2.7ⁿ + 3.5ⁿ − 5 is divisible by 24 for all n ∈ N.

Prove that n³ - 7n + 3 is divisible by 3 for all n \[\in\] N .

\[\frac{n^{11}}{11} + \frac{n^5}{5} + \frac{n^3}{3} + \frac{62}{165}n\] is a positive integer for all n ∈ N.

Let P(n) be the statement : 2ⁿ ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ∈ N?

\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\] for all n ∈ N .

x²ⁿ⁻¹ + y²ⁿ⁻¹ is divisible by x + y for all n ∈ N.

\[\text{ Prove that } \cos\alpha + \cos\left( \alpha + \beta \right) + \cos\left( \alpha + 2\beta \right) + . . . + \cos\left[ \alpha + \left( n - 1 \right)\beta \right] = \frac{\cos\left\{ \alpha + \left( \frac{n - 1}{2} \right)\beta \right\}\sin\left( \frac{n\beta}{2} \right)}{\sin\left( \frac{\beta}{2} \right)} \text{ for all n } \in N .\]

\[\text { A sequence } x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_1 = 2 \text{ and } x_k = \frac{x_{k - 1}}{k} \text{ for all natural numbers } k, k \geq 2 . \text{ Show that } x_n = \frac{2}{n!} \text{ for all } n \in N .\]

Prove by method of induction, for all n ∈ N:

1² + 2² + 3² + .... + n² = `("n"("n" + 1)(2"n" + 1))/6`

Prove by method of induction, for all n ∈ N:

(cos θ + i sin θ)ⁿ = cos (nθ) + i sin (nθ)

Prove by method of induction, for all n ∈ N:

Given that t_n+1 = 5t_n + 4, t₁ = 4, prove that t_n = 5ⁿ − 1

Answer the following:

Given that t_n+1 = 5t_n − 8, t₁ = 3, prove by method of induction that t_n = 5ⁿ⁻¹ + 2

Answer the following:

Prove by method of induction

`[(3, -4),(1, -1)]^"n" = [(2"n" + 1, -4"n"),("n", -2"n" + 1)], ∀ "n" ∈ "N"`

Answer the following:

Prove by method of induction 5²ⁿ − 2²ⁿ is divisible by 3, for all n ∈ N

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:

`(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:

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