OCR A-Level Further Maths A Series and proof: induction, standard sums, method of differences and roots of polynomials

What the series and proof strand demands

The series and proof strand of OCR A-Level Further Mathematics A (H245) is part of the mandatory Pure Core, assessed across both Pure Core papers (Y540 and Y541). It develops rigorous reasoning (induction) and the algebra of series and polynomial roots. The examiners reward complete, well-laid-out proofs, full working in summations, and careful symmetric algebra.

This guide walks through the four series and proof topics in a logical order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Proof and summation

Proof by induction establishes a statement for all integers using a base case, an inductive hypothesis, an inductive step and a concluding statement, applied to summation formulae, divisibility, recurrence relations and powers of a matrix. Summation of series uses the standard results for $\sum r$, $\sum r^2$ and $\sum r^3$, splitting a polynomial sum by linearity, factorising the result, and adjusting the limits when a sum does not start at $1$.

Differences and roots

The method of differences expresses a general term as a difference of consecutive terms (usually via partial fractions), sums by telescoping cancellation, and finds the sum to infinity where it exists. Roots of polynomials reads the sums and products of roots off the coefficients with alternating signs, evaluates symmetric functions via identities, and forms new equations whose roots are a function of the originals.

How the series and proof content is examined

A typical OCR profile for this strand:

• Induction questions. Prove a summation formula, a divisibility result, or a matrix-power formula.
• Summation questions. Sum a polynomial in $r$, or a sum with shifted limits.
• Method of differences questions. Split into partial fractions, telescope, and find the sum to infinity.
• Roots questions. Evaluate a symmetric function, or form a new equation.

Check your knowledge

A mix of recall and technique questions covering the series and proof content. Attempt them under timed conditions, then check against the solutions.

1. State the four parts of a proof by induction. (1 mark)
2. State the standard result for $\displaystyle\sum_{r=1}^{n} r^2$. (1 mark)
3. What is the key feature of a series summed by the method of differences? (1 mark)
4. For $ax^2 + bx + c = 0$ with roots $\alpha, \beta$, state $\alpha + \beta$. (1 mark)
5. State the identity for $\alpha^2 + \beta^2$ in terms of $\alpha + \beta$ and $\alpha\beta$. (1 mark)
6. To get new roots $\alpha + 3$, what substitution do you make in the original equation? (1 mark)