← Further Maths syllabus

EnglandFurther Maths

Core Pure: series and proof

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does the method of differences sum a series whose terms telescope?

The method of differences, expressing a general term as a difference of consecutive terms (often via partial fractions), summing by cancellation, and finding the sum to infinity where it exists.
A focused answer to the OCR A-Level Further Mathematics A content on the method of differences, covering how to express a general term as a difference of consecutive terms (often using partial fractions), summing the series by telescoping cancellation, writing the result in terms of n, and finding the sum to infinity when the remaining term tends to a limit.
11 min answer →

How does proof by mathematical induction work, and what must each step contain to be rigorous?

Proof by mathematical induction for summation formulae, divisibility results, recurrence relations and powers of matrices, with a correctly stated base case, inductive hypothesis, inductive step and conclusion.
A focused answer to the OCR A-Level Further Mathematics A content on proof by mathematical induction, covering the structure (base case, inductive hypothesis, inductive step and conclusion) and its use for summation formulae, divisibility results, recurrence relations and powers of a matrix, with the rigorous wording examiners require.
12 min answer →

How are the roots of a polynomial related to its coefficients, and how do you use these relationships?

The relationships between the roots and coefficients of quadratic, cubic and quartic equations, symmetric functions of the roots, and forming a new equation whose roots are a given function of the original roots.
A focused answer to the OCR A-Level Further Mathematics A content on the relationships between the roots and coefficients of polynomials, covering quadratics, cubics and quartics, the sums and products of roots, evaluating symmetric functions such as the sum of squares of the roots, and forming a new polynomial whose roots are a given function of the original roots.
11 min answer →

How do you use the standard summation formulae for sums of powers to evaluate more complicated series?

The standard results for the sum of r, r squared and r cubed, using them to sum polynomial expressions in r, splitting sums by linearity, and adjusting limits.
A focused answer to the OCR A-Level Further Mathematics A content on the summation of series, covering the standard formulae for the sum of r, r squared and r cubed, using linearity to split a sum of a polynomial in r, evaluating the resulting expression, and adjusting the limits when a sum does not start at one.
11 min answer →