--- 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. This is the common ratio r. Divide a term by its nonzero predecessor: in 9, 18, 36, 72, the quotient 18/9 is 2. Check another usable pair. A constant difference instead describes an arithmetic pattern. All-zero terms fit any supplied real ratio, so division cannot determine their ratio uniquely. **Example.** To continue a geometric sequence, find the ratio and multiply the last known term by it. In 5, 15, 45, 135, the ratio is 15/5 = 3. The next term is 135 times 3 = 405, followed by 405 times 3 = 1215. Ratios strictly between 0 and 1 shrink term magnitudes toward zero and model 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. **Example.** For first term a and ratio r, term n is a times r^(n - 1). The first term needs no multiplication, so its power is zero. For first term 4 and ratio 3, term 4 is 4 times 3^3 = 108. Substitute the term number, subtract 1 for the power, then calculate. An explicit rule gives a term directly; a recursive rule gives the first term and how to get the next. With first term 5 and each next term 3 times the previous term, the explicit rule is 5 times 3^(n - 1). Either rule can be converted to the other. To locate 1215, solve 5 times 3^(n - 1) = 1215. Dividing by 5 gives 243 = 3^5, so n - 1 = 5 and n = 6. **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)