Pure Mathematics Practice: Series and Sequences

Original practice material

These four topic-local questions total 23 marks. They combine short progression skills with convergence and linked Maclaurin reasoning suitable for Paper 1 or Paper 2 Section A.

Answers: Series and Sequences Answers.

Question 1: Arithmetic and Geometric Progressions

An arithmetic progression has first term $7$ and common difference $3$.

(a) Find its 20th term. [2]

(b) Find the sum of its first 20 terms. [3]

A geometric progression has first term $5$ and common ratio $0.8$.

(c) Explain why it has a sum to infinity and find this sum. [2]

[Total: 7]

Question 2: Sigma Notation

Evaluate

$$\sum _{r=1}^n(3r^2-2r)$$

in fully factorised form. [4]

Question 3: Convergence of a Geometric Series

The series

$$1+(2k-1)+(2k-1{)}^2+\cdots$$

has real parameter $k$.

Find the values of $k$ for which the series converges, and state its sum in terms of $k$. [5]

Question 4: Maclaurin Series and Approximation

(a) Using standard Maclaurin expansions, show that

$$e^{2x}\cos x=1+2x+\frac{3}{2}{x}^2+\frac{1}{3}{x}^3+O(x^4).$$

[5]

(b) Hence estimate $e^{0.2}\cos (0.1)$, giving your answer to 5 decimal places. [2]

[Total: 7]