Home → Curriculum Hub → Maths Lessons → Schemes of Work → A-Level Further Mathematics → Proof by Induction
This scheme of work for Core Pure 1 proof by induction builds on the sum formulae for integers, squares and cubes, and on matrix multiplication. It develops the skill of using the case \(n = k\) inside the case \(n = k + 1\). These ideas are used when a result has to be proved for every positive integer \(n\).
Students prove sum formulae first, then divisibility statements. They finish by proving a result about the powers of a matrix.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
Check the first value of \(n\). Assume the result for \(n = k\). Prove it for \(n = k + 1\) by using that assumption. If it holds at the start, and moves from \(k\) to \(k + 1\), it holds for every integer from the base case on.
Replace the sum up to \(k\) with the formula for \(k\), then rearrange until the formula for \(k + 1\) appears.
Assume \(f(k)\) is a multiple of \(m\). Write \(f(k+1)\) so that \(f(k)\) appears, then factorise \(m\) out of every term. For example,
Substitute the result for \(A^{k}\) before you multiply. The conclusion names every integer from the base case on.
| Misconception | Teaching focus |
|---|---|
| Checking the base case and then stopping. | The base case starts the chain. The inductive step is what moves from \(k\) to \(k + 1\). |
| Writing the target for \(k + 1\) and treating it as proved. | Start from the case \(k\). The algebra has to reach the statement for \(k + 1\). |
| Never substituting the statement for \(n = k\). | The result for \(n = k\) has to appear in the working for \(n = k + 1\). |
| Leaving a divisibility proof without a common factor. | Write \(f(k+1)\) so that \(f(k)\) is visible, then factorise the divisor out of every term. |
| Multiplying the matrix power without substituting the case \(k\). | \(A^{k+1}\) is \(A^{k}A\). Replace \(A^{k}\) before you multiply. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all three lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students learn to prove results about sums of series using mathematical induction.
Later, as learning progresses, students prove the sum of a cubic series using mathematical induction.
Students learn to prove results about divisibility using mathematical induction.
Later, as learning progresses, students prove statements of the form aan+b + an + c.
Students learn to prove results about matrices using mathematical induction.
Later, as learning progresses, students attempt exam-style questions on proof by mathematical induction.
A research-backed case exploring why over-reliance on automated math homework platforms and repetitive worksheets lowers student expectations, and how departments can build genuine mathematical thinking.
How to teach converting between fractions, decimals and percentages.
Four problem solving lessons to develop student’s mathematical reasoning and communication skills.
