Sequences and series practice questions (A-Level Maths)

Sample questions

The one free A-Level Maths question on this topic, from the free A-Level Maths Practice Paper. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 15 marks · Binomial expansion
   This question is about the binomial expansion of $(2 + 3x)^5$.
   $(2 + 3x)^5$
   (a)

   Find the first four terms, in ascending powers of $x$, of the binomial expansion of $(2 + 3x)^5$. Give each coefficient as an integer.

   [3]
   (b)

   Use your expansion with a suitable value of $x$ to estimate $2.03^5$, giving your answer to 4 decimal places.

   [2]
   Show answerFrom the free A-Level Maths Practice Paper →

Key terms

The definitions and formulas A-Level Maths expects for sequences and series, typeset as they appear on the paper.

Binomial expansion for a positive integer power

Every term of $(a + b)^n$ with its binomial coefficient. Watch the sign when $b$ is negative and the power when $b$ has a coefficient.

$$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$$

General binomial expansion

The infinite series for $(1 + x)^n$ when $n$ is negative or a fraction, valid only for $|x| < 1$. For $(a + bx)^n$ factor out $a^n$ first; the validity becomes $|x| < \frac{a}{b}$.

$$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots, \quad |x| < 1$$

Arithmetic sequence and series

A common difference $d$ between terms. Both forms of the sum are in the formulae booklet; the second is useful when the last term is known.

$$u_n = a + (n - 1)d, \qquad S_n = \frac{n}{2}\big(2a + (n - 1)d\big) = \frac{n}{2}(a + l)$$

Geometric sequence and series

A common ratio $r$. The sum to infinity exists only when $|r| < 1$, a condition the mark scheme expects you to state.

$$u_n = ar^{n-1}, \qquad S_n = \frac{a(1 - r^n)}{1 - r}, \qquad S_\infty = \frac{a}{1 - r}$$

Sigma notation

Compact notation for a sum of terms given by a formula in $r$. Write out the first terms to identify the type of series before using a sum formula.

$$\sum_{r=1}^{n} u_r = u_1 + u_2 + \cdots + u_n$$

Recurrence relation

Each term defined from the previous one, $u_{n+1} = f(u_n)$, with a first term given. Sequences may be increasing, decreasing or periodic; a periodic sequence repeats with a fixed order.

Frequently asked questions

Other A-Level Maths topics

• Proof, algebra and functions
• Coordinate geometry
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Integration
• Numerical methods and vectors
• Sampling and data presentation
• Probability and distributions
• Hypothesis testing
• Kinematics
• Forces, Newton's laws and moments

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