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AHL 1.15—Proof by induction, contradiction, counterexamples - IB Questionbank | RevisionDojo

Practice AHL 1.15—Proof by induction, contradiction, counterexamples with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Disprove the following statement by providing a counterexample:

"For any integers $a$ , $b$ , and $c$ , if $a$ divides $c$ and $b$ divides $c$ , then $ab$ divides $c$ ."

[3]

Question 2

HLPaper 1

Prove by contradiction that if the square of an integer is an even number, then the integer itself must be an even number.

[3]

Question 3

HLPaper 1

Disprove the following statement by providing a counterexample: "For any square matrices $A$ and $B$ of the same size, $(A+B)^2 = A^2+2AB+B^2$ ."

[4]

Question 4

HLPaper 1

Prove by mathematical induction that $\frac{d^n}{dy^n}\left(y^2e^y\right) = \left[y^2 + 2ny + n(n-1)\right]e^y$ for $n \in \mathbb{Z}^+$ .

[7]

Hence or otherwise, determine the Maclaurin series of $f(y) = y^2e^y$ in ascending powers of $y$ , up to and including the term in $y^4$ .

[3]

Question 5

HLPaper 1

Given expression $n^3 + (n+1)^3 + (n+2)^3$ .

Prove by induction that the expression is divisible by $9$ for all positive integers $n$ .

[5]

Question 6

HLPaper 1

Consider the following trigonometric expression

\sin(x+y)-\sin(x-y)

Show that the expression is equal to $2\cos(x)\sin(y)$

[3]

Hence, using mathematical induction and the above identity, prove that $\sum_{r=1}^{n} \cos((2r-1)\alpha) = \frac{\sin(2n\alpha)}{2\sin\alpha}$ for $n \in \mathbb{Z}^+$ .

[8]

Question 7

HLPaper 1

Prove by induction that if $n \in \mathbb{N}$ , $n \ge 10$ , then $2^n > n^3$ .

Hence deduce that if $n \in \mathbb{N}$ , $n \ge 10$ , then $\sqrt[3]{2} > \sqrt[n]{n}$ .

[6]

Question 8

HLPaper 1

Prove by contradiction that the equation $4x^3 + 2x^2 + 7 = 0$ has no integer roots.

[6]

Question 9

HLPaper 1

Consider the sequence defined by $a_n = 3^n - 2n$ for all positive integers $n$ .

Using proof by mathematical induction, prove that $a_n > 0$ for all integers $n \ge 1$ .

[8]

Question 10

HLPaper 1

Prove by induction that $1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$ , for $n \ge 1$ , $n \in \mathbb{N}$ .

[4]

Paper

Level

Question 1

HLPaper 1

Disprove the following statement by providing a counterexample:

"For any integers $a$ , $b$ , and $c$ , if $a$ divides $c$ and $b$ divides $c$ , then $ab$ divides $c$ ."

[3]

Question 2

HLPaper 1

Prove by contradiction that if the square of an integer is an even number, then the integer itself must be an even number.

[3]

Question 3

HLPaper 1

Disprove the following statement by providing a counterexample: "For any square matrices $A$ and $B$ of the same size, $(A+B)^2 = A^2+2AB+B^2$ ."

[4]

Question 4

HLPaper 1

Prove by mathematical induction that $\frac{d^n}{dy^n}\left(y^2e^y\right) = \left[y^2 + 2ny + n(n-1)\right]e^y$ for $n \in \mathbb{Z}^+$ .

[7]

Hence or otherwise, determine the Maclaurin series of $f(y) = y^2e^y$ in ascending powers of $y$ , up to and including the term in $y^4$ .

[3]

Question 5

HLPaper 1

Given expression $n^3 + (n+1)^3 + (n+2)^3$ .

Prove by induction that the expression is divisible by $9$ for all positive integers $n$ .

[5]

Question 6

HLPaper 1

Consider the following trigonometric expression

\sin(x+y)-\sin(x-y)

Show that the expression is equal to $2\cos(x)\sin(y)$

[3]

Hence, using mathematical induction and the above identity, prove that $\sum_{r=1}^{n} \cos((2r-1)\alpha) = \frac{\sin(2n\alpha)}{2\sin\alpha}$ for $n \in \mathbb{Z}^+$ .

[8]

Question 7

HLPaper 1

Prove by induction that if $n \in \mathbb{N}$ , $n \ge 10$ , then $2^n > n^3$ .

Hence deduce that if $n \in \mathbb{N}$ , $n \ge 10$ , then $\sqrt[3]{2} > \sqrt[n]{n}$ .

[6]

Question 8

HLPaper 1

Prove by contradiction that the equation $4x^3 + 2x^2 + 7 = 0$ has no integer roots.

[6]

Question 9

HLPaper 1

Consider the sequence defined by $a_n = 3^n - 2n$ for all positive integers $n$ .

Using proof by mathematical induction, prove that $a_n > 0$ for all integers $n \ge 1$ .

[8]

Question 10

HLPaper 1

Prove by induction that $1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$ , for $n \ge 1$ , $n \in \mathbb{N}$ .

[4]

Paper

Level

Question 1

HLPaper 1

Disprove the following statement by providing a counterexample:

"For any integers $a$ , $b$ , and $c$ , if $a$ divides $c$ and $b$ divides $c$ , then $ab$ divides $c$ ."

[3]

Question 2

HLPaper 1

Prove by contradiction that if the square of an integer is an even number, then the integer itself must be an even number.

[3]

Question 3

HLPaper 1

Disprove the following statement by providing a counterexample: "For any square matrices $A$ and $B$ of the same size, $(A+B)^2 = A^2+2AB+B^2$ ."

[4]

Question 4

HLPaper 1

Prove by mathematical induction that $\frac{d^n}{dy^n}\left(y^2e^y\right) = \left[y^2 + 2ny + n(n-1)\right]e^y$ for $n \in \mathbb{Z}^+$ .

[7]

Hence or otherwise, determine the Maclaurin series of $f(y) = y^2e^y$ in ascending powers of $y$ , up to and including the term in $y^4$ .

[3]

Question 5

HLPaper 1

Given expression $n^3 + (n+1)^3 + (n+2)^3$ .

Prove by induction that the expression is divisible by $9$ for all positive integers $n$ .

[5]

Question 6

HLPaper 1

Consider the following trigonometric expression

\sin(x+y)-\sin(x-y)

Show that the expression is equal to $2\cos(x)\sin(y)$

[3]

Hence, using mathematical induction and the above identity, prove that $\sum_{r=1}^{n} \cos((2r-1)\alpha) = \frac{\sin(2n\alpha)}{2\sin\alpha}$ for $n \in \mathbb{Z}^+$ .

[8]

Question 7

HLPaper 1

Prove by induction that if $n \in \mathbb{N}$ , $n \ge 10$ , then $2^n > n^3$ .

Hence deduce that if $n \in \mathbb{N}$ , $n \ge 10$ , then $\sqrt[3]{2} > \sqrt[n]{n}$ .

[6]

Question 8

HLPaper 1

Prove by contradiction that the equation $4x^3 + 2x^2 + 7 = 0$ has no integer roots.

[6]

Question 9

HLPaper 1

Consider the sequence defined by $a_n = 3^n - 2n$ for all positive integers $n$ .

Using proof by mathematical induction, prove that $a_n > 0$ for all integers $n \ge 1$ .

[8]

Question 10

HLPaper 1

Prove by induction that $1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$ , for $n \ge 1$ , $n \in \mathbb{N}$ .

[4]

AHL 1.15—Proof by induction, contradiction, counterexamples Questionbank