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Questions >> CBSE XI >> Math >> Principle of Mathematical Induction

Prove the following statement by the principle of mathematical induction : $1+5+9+...+(4n-3)=n(2n-1)$ for all natural numbers $n$ .

answered Feb 3, 2021 by abdulrahimshafiuyusuf

1 answer

Choose the correct answer for all $n \in N, 3.5^{2n+1}+2^{3n+1}$ is divisible by

answered Sep 25, 2020 by apoorva14544

1 answer

If $P(n) : 2n < n!, n \in N$ , then $P(n)$ is true for all $n \geq$ ______ .

answered Aug 14, 2019 by s9466813326

1 answer

State whether the following statement is true or false. Justify. 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 $n \in N$ .

asked May 6, 2014 by thanvigandhi_1

0 answers

Choose the correct answer if $x^n-1$ is divisible by $x - k$ , then the least positive integral value of $k$ is

asked May 6, 2014 by thanvigandhi_1

0 answers

Choose the correct answer if $10^n+3.4^{n+2}+k$ is divisible by 9 for all $n \in N$ , then the least positive integral value of $k$ is

asked May 6, 2014 by thanvigandhi_1

0 answers

Prove that number of subsets of a set containing $n$ distinct elements is $2^n$ , for all $n \in N$ by using principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

Prove that, $\large\frac{1}{n+1}$ $+ \large\frac{1}{n+2}$ $+...+\large\frac{1}{2n}$ $> \large\frac{13}{24}$ , for all natural numbers $n > 1$ by using principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

Show that $\large\frac{n^5}{5}$ $+\large\frac{n^3}{3}$ $+\large\frac{7n}{15}$ is a natural number for all $n \in N$ by using the principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

Prove that, $\sin \theta + \sin 2\theta + \sin 3\theta+...+ \sin n\theta = \large\frac{\Large\frac{\sin n \theta}{2} \sin \Large\frac{(n+1)}{2} \theta}{\sin \Large\frac{\theta}{2}}$ , for all $n \in N$ by using principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

Prove that, $\cos \theta, \cos 2\theta \cos 2^2\theta....\cos 2^{n-1} \theta = \large\frac{\sin 2^n \theta}{2^n \sin \theta}$ , for all $n \in N$ by using the principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

Prove that for all $n \in N$ $\cos \alpha+ \cos ( \alpha + \beta ) + \cos ( \alpha + 2 \beta )+...+ \cos ( \alpha + (n-1) \beta )$ $= \large\frac{ \cos \bigg( \alpha + \bigg( \Large\frac{n-1}{2} \bigg) \beta \bigg) \sin \bigg( \Large\frac{n \beta}{2} \bigg)}{\sin \Large\frac{\beta}{2}}$ by using principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

A sequence $d_1, d_2, d_3...$ is defined by letting $d_1=2$ and $d_k=\large\frac{d_{k-1}}{k}$ for all natural numbers $k \geq 2$ . Show that $d_n = \large\frac{2}{n!}$ for all $n \in N$ by using principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

A sequence $b_0, b_1, b_2...$ 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.

asked May 6, 2014 by thanvigandhi_1

0 answers

A sequence $a_1, a_2, a_3...$ is defined by letting $a_1=3$ and $a_k=7a_{k-1}$ for all natural numbers $k \geq 2$ . Show that $a_n = 3.7^{n-1}$ for all natural numbers by using principle of mathematical induction.

asked May 6, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $1+2+2^2+...+2^n=2^{n+1}-1$ for all natural numbers $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $2+4+6+...+2n=n^2+n$ for all natural numbers $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $\sqrt n <\large\frac{1}{\sqrt 1}$ $+\large\frac{1}{\sqrt 2}$ $+...+\large\frac{1}{\sqrt n}$ , for all natural numbers $n \geq 2$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $2n < (n+2)!$ for all natural number $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $n^2 < 2^n$ for all natural numbers $n \geq 5$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $n(n^2+5)$ is divisible by 6, for each natural number $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $n^3-n$ is divisible by 6, for each natural number $n \geq 2$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : For any natural number $n$ , $x^n-y^n$ is divisible by $x-y$ , where $x$ and $y$ are any integers with $x \neq y$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : For any natural number $n$ , $7^{n}-2^n$ is divisible by 5.

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $3^{2n}-1$ is divisible by 8, for all natural numbers $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $n^{3}-7n+3$ is divisible by 3, for all natural numbers $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $2^{3n}-1$ is divisible by 7, for all natural numbers $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the following statement by the principle of mathematical induction : $4^n-1$ is divisible by 3, for each natural numbers $n$ .

asked May 5, 2014 by thanvigandhi_1

0 answers

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

asked May 5, 2014 by thanvigandhi_1

0 answers

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

asked May 5, 2014 by thanvigandhi_1

0 answers

Prove the rule of exponents $(ab)^n=a^nb^n$ by using principle of mathematical induction for every natural number.

answered May 5, 2014 by thanvigandhi_1

1 answer

Prove that $1^2+2^2+...+n^2 > \large\frac{n^3}{3}$ $, n \in N$

answered May 5, 2014 by thanvigandhi_1

1 answer

Prove that $2.7^n+3.5^n-5$ is divisible by 24, for all $n \in N$ .

answered May 3, 2014 by thanvigandhi_1

1 answer

Prove that $(1+x)^n \geq (1+nx)$ , for all natural number $n$ , where $x > -1$ .

answered May 3, 2014 by thanvigandhi_1

1 answer

For every positive integer $n$ , prove that $7^n-3^n$ is divisible by 4.

answered May 3, 2014 by thanvigandhi_1

1 answer

For all $n \geq 1$ , prove that $\large\frac{1}{1.2}$ $+\large\frac{1}{2.3}$ $+\large\frac{1}{3.4}$ $+...+\large\frac{1}{n(n+1)}$ $=\large\frac{n}{(n+1)}$ .

answered May 2, 2014 by thanvigandhi_1

1 answer

Prove that $2^n > n$ for all positive integers $n$ .

answered May 2, 2014 by thanvigandhi_1

1 answer

For all $n \geq 1$ , prove that $1^2+2^2+3^2+4^4+...+n^2=\large\frac{n(n+1)(2n+1)}{6}$ .

answered May 2, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $(2n+7) < (n+3)^2$

answered May 1, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $41^n-14^n$ is a multiple of 27.

answered May 1, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $3^{2n+2}-8n-9$ is divisible by 8.

answered May 1, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $x^{2n}-y^{2n}$ is divisible by $x+y$ .

answered May 1, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $10^{2n-1}+1$ is divisible by 11.

answered May 1, 2014 by thanvigandhi_1

1 answer

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

answered May 1, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all n ∈ N: $1+2+3+...+n<\large\frac{1}{8}$ $(2n+1)^2$

answered May 1, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $\large\frac{1}{3.5}$ $+\large\frac{1}{5.7}$ $+\large\frac{1}{7.9}$ $+...+\large\frac{1}{(2n+1)(2n+3)}$ $=\large\frac{n}{3(2n+3)}$

answered Apr 30, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $\large\frac{1}{1.4}$ $+\large\frac{1}{4.7}$ $+\large\frac{1}{7.10}$ $+...+\large\frac{1}{(3n-2)(3n+1)}$ $=\large\frac{n}{(3n+1)}$

answered Apr 30, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $1^2+3^2+5^2+...+(2n-1)^2=\large\frac{n(2n-1)(2n+1)}{3}$

answered Apr 30, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $\bigg( 1+\large\frac{1}{1} \bigg)$ $\bigg( 1+\large\frac{1}{2} \bigg)$ $\bigg( 1+\large\frac{1}{3} \bigg)$ $...\bigg( 1+\large\frac{1}{n} \bigg)$ $=(n+1)$

answered Apr 30, 2014 by thanvigandhi_1

1 answer

Prove the following by using the principle of mathematical induction for all $n \in N$ $\bigg( 1+ \large\frac{3}{1} \bigg)$ $\bigg( 1+ \large\frac{5}{4} \bigg)$ $\bigg( 1+ \large\frac{7}{9} \bigg)$ $...\bigg( 1+ \large\frac{(2n+1)}{n^2} \bigg)$ $=(n+1)^2$

answered Apr 30, 2014 by thanvigandhi_1

1 answer

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