Proof by Mathematical Induction

We estimate ME12-PI is worth about 3 marks out of 100 under the new Mathematics Extension 1 syllabus. Practise 5 exam-style questions (short answer, multiple choice) with instant marking and a worked solution for every question.

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Example questions

Example 1 · 1 mark · multiple choice

A student 'proves' that $3^{n} + 1$ is divisible by 4 for all $n \geq 1$ . They show: if $3^{k} + 1 = 4 M$ , then $3^{k + 1} + 1 = 3 (4 M - 1) + 1 = 12 M - 2$ , and claim this is divisible by 4. Which statement is correct?

AThe result is true but needs a different inductive step
BThe proof is correct
CThe claim that $12 M - 2$ is divisible by 4 is false, and the result itself is false (for example when $n = 2$ )
DOnly the base case is missing

Example 2 · 3 marks · short answer

Use mathematical induction to prove that $1 + 3 + 5 + \dots + (2 n - 1) = n^{2}$ for all integers $n \geq 1$ .

Example 3 · 4 marks · short answer

Use mathematical induction to prove that $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \dots + \frac{1}{n ( n + 1 )} = \frac{n}{n + 1}$ for all integers $n \geq 1$ .

Answers and worked solutions appear when you practise these questions.

Proof by Mathematical Induction

Example questions

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