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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Pair first with last and multiply, and let sigma name the start, stop, and pattern.
2 · Watch
Take it off screen
Where it sits
Learn first
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.