IGCSE · Practice questions · Sequences and the nth term

Find the formula.

Six original Cambridge-style questions on sequences: continuing patterns, term-to-term rules, the nth term of linear sequences and using it to answer questions.

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

Difference first, then adjust.

Differences: subtract each term from the next. Constant means linear.
Start the formula: the common difference \(d\) goes in front of \(n\).
Adjust: add or subtract so that \(n = 1\) gives the first term.
Check: test \(n = 2\) as well.
Use it: set the formula equal to a number; a whole-number \(n\) means it is a term.

Recall

[2 marks]

Write down the next two terms and the term-to-term rule: \(31, 25, 19, 13, \ldots\)

Rule: subtract 6. ✓
Next terms: 7, 1. ✓

Recall

[2 marks]

Write down the next two terms: \(2, 3, 5, 8, 12, \ldots\)

Differences: 1, 2, 3, 4, so the next differences are 5 and 6. ✓
\(12 + 5 = 17\), \(17 + 6 = 23\). ✓

Calculation

[3 marks]

(a) Find the nth term of \(9, 16, 23, 30, \ldots\) (b) Find the 50th term.

(a) Difference 7, so \(7n\); \(7(1) = 7\), first term 9, so \(7n + 2\). ✓
(b) \(7(50) + 2 = 352\). ✓

Calculation

[2 marks]

The nth term of a sequence is \(50 - 3n\). Find the first term that is negative.

\(50 - 3n < 0\) gives \(3n > 50\), so \(n > 16.67\) and \(n = 17\). ✓
\(T_{17} = 50 - 51 = -1\). ✓

Analysis

[3 marks]

A sequence begins \(5, 12, 19, 26, \ldots\) Is 245 a term of this sequence? Show your working.

Difference 7, so the nth term is \(7n - 2\). ✓
\(7n - 2 = 245\) gives \(7n = 247\), so \(n = 35.29...\) ✓
\(n\) is not a whole number, so 245 is NOT a term. ✓

Analysis

[5 marks]

Rectangular tables are pushed together in a row. One table seats 6 people, two tables seat 10 and three tables seat 14.

(a) Find a formula for the number of seats with \(n\) tables. (b) How many seats are there with 12 tables? (c) 75 people come to a party. What is the smallest number of tables needed? (d) Explain why the number of seats is always even.

(a) Difference 4, first term 6, so \(4n + 2\). ✓
(b) \(4(12) + 2 = 50\). ✓
(c) \(4n + 2 \ge 75\), so \(4n \ge 73\) and \(n \ge 18.25\). Round UP: 19 tables (18 tables seat only 74). ✓
(d) \(4n + 2 = 2(2n + 1)\), which is 2 times a whole number, so it is always even. ✓

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.