---
title: "Geometric Sequences and Common Ratios"
description: "Recognise a sequence built by multiplying by the same number each time, find the common ratio and the nth term rule, and link it to growth and decay."
canonical: https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY
source: https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY.md
retrieved: 2026-09-12
---

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# Geometric Sequences and Common Ratios

Recognise a sequence built by multiplying by the same number each time, find the common ratio and the nth term rule, and link it to growth and decay.

Subject: Mathematics · Area: Algebra · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY

## Ready when they can

- Find the common ratio by dividing consecutive terms
- Write the nth term as a first term times r to a power
- Continue a decaying sequence with a ratio between 0 and 1

## Lesson: Lists that multiply each step

A geometric sequence multiplies by the same number each step, and that multiplier is the common ratio. Find it by dividing any term by the one before: in 3, 6, 12, 24, 6 divided by 3 is 2. Check divisions, not differences: a constant difference means an arithmetic pattern instead.

**Example.** 5, 15, 45, 135 grows by times 3 each time, so the next two terms are 405 then 1215. The nth term rule is first term times ratio to n minus 1: 5 times 3^(n-1). The power is n minus 1 because the first term needs zero multiplications.

Ratios below 1 shrink the terms toward zero, and that links lists to decay. 100, 50, 25 continues to 12.5 then 6.25 with ratio 0.5. Track the term number carefully: mistaking it for the power is the classic error.

**Tip.** To jump ahead, write the rule before raising anything high. First term 4, ratio 3: term 4 = 4 times 3^3 = 108. Two written terms confirm the ratio before the big powers arrive.

**Recap.** Divide neighbours for the ratio, then first term times ratio to n minus 1.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Nth-Term Rules](https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO)
- [Compound Growth and Decay](https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf)

## Opens up

- [Geometric Series and the Sum to Infinity](https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS)
- [Big-O Notation and Orders of Growth](https://lightmysky.com/learn/computing/big-o-notation-and-orders-of-growth-mt_mkqgxhJ509)
- [Sequences and Their Limits](https://lightmysky.com/learn/mathematics/sequences-and-their-limits-mt_PQl3Q6n5dc)