--- title: "Arithmetic Series and Sigma Notation" description: "Add the terms of a linear sequence with the pairing argument behind Sₙ = n(a + l)/2, and read and write sums in sigma notation." canonical: https://lightmysky.com/learn/mathematics/arithmetic-series-and-sigma-notation-mt_bAIPng87JO source: https://lightmysky.com/learn/mathematics/arithmetic-series-and-sigma-notation-mt_bAIPng87JO.md retrieved: 2026-09-12 --- > **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`. # Arithmetic Series and Sigma Notation Add the terms of a linear sequence with the pairing argument behind Sₙ = n(a + l)/2, and read and write sums in sigma notation. Subject: Mathematics · Area: Algebra · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/arithmetic-series-and-sigma-notation-mt_bAIPng87JO ## Ready when they can - Find the sum of the first 50 terms of an arithmetic sequence - Explain the pairing argument that produces the formula - Write a given sum in sigma notation and evaluate it ## Lesson: Adding arithmetic runs with sigma An arithmetic sequence grows by the same step each time, like 3, 7, 11 with a step of 4. From the first term and the step you can jump to any term without listing them all. The sigma sign is tidy shorthand for adding a run of terms: below it sits the starting counter, above it the stopping counter, and beside it the pattern for each term. **Example.** Pair terms to add fast. Write 1 to 100 forwards and backwards and every column adds to 101, so 50 columns give 5050. In general multiply the count of terms by the average of first and last: S = n(a + l)/2. For 1 to 50 that is 25 pairs of 51, which is 1275. For the first 50 odd numbers each pair makes 100, so 25 pairs give 2500. **Example.** Try 3, 7, 11 and add its first 20 terms. The step is 4, so the 20th term is 3 + 19 times 4 = 79. Pair first with last: 3 + 79 = 82, and 10 such pairs give 820. The first 100 even numbers are twice 1 to 100, so 2 times 5050 = 10100. Read sigma by testing small values of the counter. The sum of (2k + 1) for k = 1 to 4 gives 3, 5, 7, 9, which add to 24. To write 5 + 7 + 9 + 11 + 13 in sigma, test (2k + 3): k = 1 gives 5 and k = 5 gives 13, so it is the sum of (2k + 3) for k = 1 to 5. Check any sigma sum by writing a few terms by hand. **Recap.** Pair first with last and multiply, and let sigma name the start, stop, and pattern. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Nth-Term Rules](https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO) ## Opens up - [Proof by Induction and Strong Induction](https://lightmysky.com/learn/mathematics/proof-by-induction-and-strong-induction-mt_7lvC02JBOC) - [Geometric Series and the Sum to Infinity](https://lightmysky.com/learn/mathematics/geometric-series-and-the-sum-to-infinity-mt_HHxFOv6vXS)