IGCSE · Practice questions · Arithmetic progressions

Find a and d, then use them.

Six original Cambridge-style questions on arithmetic progressions: recognising an AP, the \(n\)th term, the sum of \(n\) terms, finding \(a\) and \(d\), the least number of terms for a total, a context and three terms in an unknown, all without a calculator.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Name \(a\), \(d\) and \(n\), then pick a formula.

Recognise: constant difference means an AP; constant ratio means a GP.
One term: \(u_{n} = a + (n - 1)d\). Two facts give two equations in \(a\) and \(d\); solve them simultaneously.
A total: \(S_{n} = \frac{n}{2}(a + l)\) if you know the last term, \(\frac{n}{2}\big(2a + (n - 1)d\big)\) if you know \(d\).
Count terms with \(n = \frac{l - a}{d} + 1\).
Least \(n\): form the inequality, then test the whole numbers either side and round UP.
Three terms: middle term \(\times 2\) = first + third.

Recall

[3 marks]

For each sequence, state whether it is arithmetic, geometric or neither. Give the common difference or the common ratio where there is one.

(a) 40, 33, 26, 19, ...

(b) 5, 15, 45, 135, ...

(a) Differences \(-7, -7, -7\): arithmetic, \(d = -7\). ✓
(b) Ratios \(3, 3, 3\) (differences 10, 30, 90 are not equal): geometric, ratio 3. ✓
(c) Differences 1, 2, 3 and ratios \(\frac{3}{2}, \frac{5}{3}, \frac{8}{5}\) are not constant: neither. ✓

Calculation

[4 marks]

The 4th term of an arithmetic progression is 23 and the 9th term is 48.

(a) Find the first term and the common difference.

(b) Find the sum of the first 20 terms.

(a) \(a + 3d = 23\) and \(a + 8d = 48\). Subtract: \(5d = 25\), so \(d = 5\). ✓
\(a = 23 - 15 = 8\). ✓
(b) \(S_{20} = \dfrac{20}{2}\big(2(8) + 19(5)\big) = 10(16 + 95)\). ✓
\(= 10 \times 111 = 1110\). ✓

Calculation

[3 marks]

Find the sum of all the integers between 100 and 400 that are multiples of 9.

First: \(9 \times 12 = 108\). Last: \(9 \times 44 = 396\). ✓
Number of terms: \(44 - 12 + 1 = 33\). ✓
\(S_{33} = \dfrac{33}{2}(108 + 396) = \dfrac{33}{2} \times 504 = 33 \times 252 = 8316\). ✓

Analysis

[6 marks]

The third term of an arithmetic progression is 16 and the sum of the first 12 terms is 318.

(a) Find the first term and the common difference.

(b) Find the least number of terms for which the sum of the progression exceeds 1000.

(a) \(a + 2d = 16\) and \(\dfrac{12}{2}(2a + 11d) = 318\), so \(2a + 11d = 53\). ✓
Doubling the first: \(2a + 4d = 32\). Subtract: \(7d = 21\), so \(d = 3\) and \(a = 16 - 6 = 10\). ✓
(b) \(S_{n} = \dfrac{n}{2}\big(20 + 3(n - 1)\big) = \dfrac{n(3n + 17)}{2}\). Need \(n(3n + 17) > 2000\). ✓
Test whole numbers: \(n = 20\) gives \(20 \times 77 = 1540\), too small. \(n = 23\) gives \(23 \times 86 = 1978 < 2000\), that is \(S_{23} = 989\). ✓
\(n = 24\) gives \(24 \times 89 = 2136 > 2000\), that is \(S_{24} = 1068 > 1000\). ✓
The least number of terms is 24. ✓

Analysis

[4 marks]

A theatre has 25 rows of seats. The front row has 18 seats and each row has 2 more seats than the row in front of it.

(a) Find the number of seats in the back row.

(b) Find the total number of seats in the theatre.

The rows form an AP with \(a = 18\), \(d = 2\), \(n = 25\). ✓
(a) \(u_{25} = 18 + 24(2) = 66\) seats. ✓
(b) \(S_{25} = \dfrac{25}{2}(18 + 66) = \dfrac{25}{2} \times 84\). ✓
\(= 25 \times 42 = 1050\) seats. ✓

Analysis

[5 marks]

The first three terms of an arithmetic progression are \(k + 2\), \(3k\) and \(4k + 1\).

(a) Find the value of \(k\) and the common difference.

(b) Find the 20th term and the sum of the first 20 terms.

(a) The middle term is the average of the other two: \(2(3k) = (k + 2) + (4k + 1)\). ✓
\(6k = 5k + 3\), so \(k = 3\). The terms are 5, 9, 13, so \(d = 4\). ✓
(b) \(u_{20} = 5 + 19(4) = 81\). ✓
\(S_{20} = \dfrac{20}{2}(5 + 81) = 10 \times 86 = 860\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.