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

प्रश्न

प्रमेय

उत्तर

Let P(n) be the given statement, that is

P(n) : 1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

Note that P(1) is true.

Since P(1): 1 × 1! = 1

= 2 – 1

= 2! – 1.

Assume that P(n) is true for some natural number k.

i.e., P(k) : 1 × 1! + 2 × 2! + 3 × 3! + ... + k × k! = (k + 1)! – 1

To prove P(k + 1) is true.

We have P(k + 1) : 1 × 1! + 2 × 2! + 3 × 3! + ... + k × k! + (k + 1) × (k + 1)!,

= (k + 1)! – 1 + (k + 1)! × (k + 1)

= (k + 1 + 1) (k + 1)! – 1

= (k + 2) (k + 1)! – 1 = ((k + 2)! – 1)

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

Therefore, by the Principle of Mathematical Induction, P(n) is true for all natural number n.

shaalaa.com

Principle of Mathematical Induction

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 9 Q 8 Q 10

अध्याय 4: Principle of Mathematical Induction - Solved Examples [पृष्ठ ६७]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 11

अध्याय 4 Principle of Mathematical Induction
Solved Examples | Q 9 | पृष्ठ ६७

वीडियो ट्यूटोरियलVIEW ALL [1]

view
वीडियो ट्यूटोरियल For All Subjects
Principle of Mathematical Induction

video tutorial00:17:42

संबंधित प्रश्न

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

`1^3 + 2^3 + 3^3 + ... + n^3 = ((n(n+1))/2)^2`

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

`1/3.5 + 1/5.7 + 1/7.9 + ...+ 1/((2n + 1)(2n +3)) = n/(3(2n +3))`

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

Given an example of a statement P (n) such that it is true for all n ∈ N.

\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . . + \frac{1}{(3n - 2)(3n + 1)} = \frac{n}{3n + 1}\]

1² + 3² + 5² + ... + (2n − 1)² = \[\frac{1}{3}n(4 n^2 - 1)\]

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

Prove that 1 + 2 + 2² + ... + 2ⁿ = 2ⁿ⁺¹ - 1 for all n \[\in\] N .

\[\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x\] for all n ∈ N and \[0 < x < \frac{\pi}{2}\]

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 .

\[\text{ Prove that } \frac{1}{n + 1} + \frac{1}{n + 2} + . . . + \frac{1}{2n} > \frac{13}{24}, \text{ for all natural numbers } n > 1 .\]

Show by the Principle of Mathematical induction that the sum S_n of then terms of the series \[1^2 + 2 \times 2^2 + 3^2 + 2 \times 4^2 + 5^2 + 2 \times 6^2 + 7^2 + . . .\] is given by \[S_n = \binom{\frac{n \left( n + 1 \right)^2}{2}, \text{ if n is even} }{\frac{n^2 \left( n + 1 \right)}{2}, \text{ if n is odd } }\]

\[\text{ Using principle of mathematical induction, prove that } \sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + . . . + \frac{1}{\sqrt{n}} \text{ for all natural numbers } n \geq 2 .\]

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:

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

1² + 4² + 7² + ... + (3n − 2)² = `"n"/2 (6"n"^2 - 3"n" - 1)`

Answer the following:

Prove by method of induction log_a xⁿ = n log_ax, x > 0, n ∈ N

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.

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. Justify your answer.