(ab)n = anbn for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
n(n + 1) (n + 5) is a multiple of 3 for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
72n + 23n−3. 3n−1 is divisible by 25 for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
2.7n + 3.5n − 5 is divisible by 24 for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
11n+2 + 122n+1 is divisible by 133 for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
Given \[a_1 = \frac{1}{2}\left( a_0 + \frac{A}{a_0} \right), a_2 = \frac{1}{2}\left( a_1 + \frac{A}{a_1} \right) \text{ and } a_{n + 1} = \frac{1}{2}\left( a_n + \frac{A}{a_n} \right)\] for n ≥ 2, where a > 0, A > 0.
Prove that \[\frac{a_n - \sqrt{A}}{a_n + \sqrt{A}} = \left( \frac{a_1 - \sqrt{A}}{a_1 + \sqrt{A}} \right) 2^{n - 1}\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
Prove that n3 - 7n + 3 is divisible by 3 for all n \[\in\] N .
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all n \[\in\] N .
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
7 + 77 + 777 + ... + 777 \[{. . . . . . . . . . .}_{n - \text{ digits } } 7 = \frac{7}{81}( {10}^{n + 1} - 9n - 10)\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\frac{n^7}{7} + \frac{n^5}{5} + \frac{n^3}{3} + \frac{n^2}{2} - \frac{37}{210}n\] is a positive integer for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\frac{n^{11}}{11} + \frac{n^5}{5} + \frac{n^3}{3} + \frac{62}{165}n\] is a positive integer for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x\] for all n ∈ N and \[0 < x < \frac{\pi}{2}\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
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?
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\] for all n ∈ N .
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + . . . + \frac{1}{n^2} < 2 - \frac{1}{n}\] for all n ≥ 2, n ∈ N
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
x2n−1 + y2n−1 is divisible by x + y for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\sin x + \sin 3x + . . . + \sin (2n - 1)x = \frac{\sin^2 nx}{\sin x}\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\text{ Prove that } \cos\alpha + \cos\left( \alpha + \beta \right) + \cos\left( \alpha + 2\beta \right) + . . . + \cos\left[ \alpha + \left( n - 1 \right)\beta \right] = \frac{\cos\left\{ \alpha + \left( \frac{n - 1}{2} \right)\beta \right\}\sin\left( \frac{n\beta}{2} \right)}{\sin\left( \frac{\beta}{2} \right)} \text{ for all n } \in N .\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\text{ Prove that } \frac{1}{n + 1} + \frac{1}{n + 2} + . . . + \frac{1}{2n} > \frac{13}{24}, \text{ for all natural numbers } n > 1 .\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\text{ Given } a_1 = \frac{1}{2}\left( a_0 + \frac{A}{a_0} \right), a_2 = \frac{1}{2}\left( a_1 + \frac{A}{a_1} \right) \text{ and } a_{n + 1} = \frac{1}{2}\left( a_n + \frac{A}{a_n} \right) \text{ for } n \geq 2, \text{ where } a > 0, A > 0 . \]
\[\text{ Prove that } \frac{a_n - \sqrt{A}}{a_n + \sqrt{A}} = \left( \frac{a_1 - \sqrt{A}}{a_1 + \sqrt{A}} \right) 2^{n - 1} .\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
