# Arithmetic and geometric sequences · Years 11 and 13

> Explanation, worked examples and free practice for the national curriculum in England.

In Arithmetic and geometric sequences (Years 11 and 13) you practise 8 skills: The rule of a sequence: the general term and a recurrence · Arithmetic sequences: the general term · The sum of an arithmetic sequence · Geometric sequences: the general term · The sum of a geometric sequence · Infinite geometric sequences with a finite sum · Properties of sequences · A sequence on a graph.

A sequence is a list of numbers in order. It can be given by a formula for the general term, which gives each term from its place, or by a recurrence, which gives each term from the one before it.

In an arithmetic sequence the difference between each term and the one before it is fixed: it is called the common difference. To reach the term in a given place, add the common difference to the first term one time fewer than the place number. The sum of the terms up to a given place is the first plus the last, times the number of terms, divided by two.

In a geometric sequence each term divided by the one before it gives the same number: it is called the common ratio. To reach the term in a given place, multiply the first term by the ratio one time fewer than the place number. When the absolute value of the ratio is less than one, the sum of all the terms of the infinite sequence has a finite value: the first term divided by one minus the ratio.

**Tip:** Before working anything out, check whether each term is the one before plus a fixed amount, or times a fixed number.

## What you learn

### The rule of a sequence: the general term and a recurrence

- Example: A sequence is defined by a₁ = −7 and aₙ₊₁ = aₙ + 5 . What is its fourth term?
- Answer: 8
- Solution: One after another: a₂ = −7 + 5 = −2 , a₃ = −2 + 5 = 3 , a₄ = 3 + 5 = 8 .

### Arithmetic sequences: the general term

- Example: An arithmetic sequence has a₁ = 9 and d = −4 . What is a₂₀ ?
- Answer: −67
- Solution: By the formula a₂₀ = 9 + 19 × (−4) = −67 .

### The sum of an arithmetic sequence

- Example: An arithmetic sequence has a₁ = −8 and d = 5 . What is S₉ , the sum of the first 9 terms?
- Answer: 108
- Solution: The last term: a₉ = −8 + 8 × 5 = 32 . So a₁ + a₉ = 24 , and the sum is S₉ = 24 × 9 ÷ 2 = 108 .

### Geometric sequences: the general term

- Example: A geometric sequence has a₁ = 3 and q = 2 . What is a₆ ?
- Answer: 96
- Solution: By the formula a₆ = 3 × 2⁵ , and 2⁵ = 32 . So a₆ = 96 .

### The sum of a geometric sequence

- Example: The first three terms of a geometric sequence are 1, 2, 4 . What is S₉ , the sum of the first 9 terms?
- Answer: 511
- Solution: Here a₁ = 1 and q = 2 , so q − 1 = 1 . The sum: S₉ = 1 × (2⁹ − 1) = 1 × 511 = 511 .

### Infinite geometric sequences with a finite sum

- Example: An infinite geometric sequence has a₁ = −9 and q = −1 . Does the sum of all its terms have a finite value?
- Options: Yes, since |q| < 1 · No, since |q| ≥ 1
- Answer: No, since |q| ≥ 1
- Solution: Here |q| = 1 , which is not less than one. No, since |q| ≥ 1 : the terms do not shrink towards zero, so the sum has no finite value.

### Properties of sequences

- Example: The sequence aₙ = 4 + n is an arithmetic sequence. Which of these sequences is arithmetic too?
- Options: aₙ² · 1 ÷ aₙ · 2^(aₙ) · aₙ + 3
- Answer: aₙ + 3
- Solution: In aₙ + 3 each term is 3 more than the original, so the difference between neighbours stays 1 . In aₙ² the first terms are 25, 36, 49 , and the differences 11 and 13 are not equal. In 1 ÷ aₙ the first terms are 1/5, 1/6, 1/7 , and the differences −1/30 and −1/42 are not equal. In 2^(aₙ) the first terms are 32, 64, 128 , and the differences 32 and 64 are not equal.

### A sequence on a graph

- Example: The graph shows the first six terms of an arithmetic sequence as the points (n, aₙ) . What is the common difference? (Six points on a grid, at the places one to six, with the heights −4, −1, 2, 5, 8, 11)
- Answer: 3
- Solution: From the graph a₁ = −4 and a₂ = −1 , so d = −1 − (−4) = 3 . Each point is higher than the one before by the same amount.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4) and selected AS and A level mathematics practice in England (the GCE subject content).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 11 topics](https://kixplo.com/uk/en/maths/year-11/)