IB Mathematics AA HL  Popular Quizzes
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Question 1
[Maximum mark: 6]

Show that $(2n1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]

Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
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Question 2
[Maximum mark: 4]
Using the method of proof by contradiction, prove that $\sqrt{3}$ is irrational.
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Question 3
[Maximum mark: 6]
Let $r \in \mathbb{R}, r\neq 1$. Use the method of mathematical induction to prove that
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Question 4
[Maximum mark: 7]
The Fibonacci sequence is defined as follows:
Prove by mathematical induction that $a_1^2+a_2^2+\cdots+a_n^2=a_na_{n+1}$, where $n\in\mathbb{Z}^+$.
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Question 5
[Maximum mark: 6]
Using the principle of mathematical induction, prove that $n(n^2+5)$ is divisible by $6$ for all integers $n \geq 1$.
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Question 6
[Maximum mark: 8]

Solve the inequality $x^2 \geq 2x + 3$. [2]

Use mathematical induction to prove that $2^n > n^2  2$ for all $n \in \mathbb{Z}^+$, $n \geq 3$.[6]
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Question 7
[Maximum mark: 14]

Show that $\dfrac{1}{2\sqrt{n+1}} < \sqrt{n+1}  \sqrt{n}$, where $n \in \mathbb{Z},\hspace{0.1em} n\geq 0$. [3]

Hence show that $\dfrac{1}{\sqrt{2}} < 2\sqrt{2}  2$. [2]

Prove by mathematical induction that
$\hspace{4em} \sum_{r = 2}^n \dfrac{1}{\sqrt{r}} < 2\sqrt{n}  2 \hspace{2em} \text{for all $n \in \mathbb{Z}^+$, $n \geq 2$.}$[9]
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