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Recent questions tagged mathematical-induction

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

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

asked May 6, 2014 by thanvigandhi_1

1 answer

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

asked May 6, 2014 by thanvigandhi_1

1 answer

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 : \[\] $1+5+9+...+(4n-3)=n(2n-1)$ for all natural numbers $n$.

asked May 5, 2014 by thanvigandhi_1

1 answer

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.

asked May 5, 2014 by thanvigandhi_1

1 answer

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

asked May 5, 2014 by thanvigandhi_1

1 answer

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

asked May 3, 2014 by thanvigandhi_1

1 answer

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

asked May 3, 2014 by thanvigandhi_1

1 answer

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

asked 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)}$.

asked May 2, 2014 by thanvigandhi_1

1 answer

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

asked 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}$.

asked 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$

asked 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.

asked 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.

asked 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 $x+y$.

asked 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.

asked 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.

asked 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$

asked 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)}$

asked 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)}$

asked 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}$

asked 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)$

asked 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$

asked Apr 30, 2014 by thanvigandhi_1

1 answer

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