Arithmetic and Geometric Series
A free Algebra II lesson from the “Sequences, Series, and Counting” unit, with a worked example and practice problems including step-by-step solutions.
A series is the sum of the terms of a sequence. For arithmetic series with first term a_1, last term a_n, and n terms: S_n = (n/2)(a_1 + a_n). For geometric series with first term a_1 and common ratio r: S_n = a_1 * (1 - r^n) / (1 - r). When |r| < 1 the geometric series has an infinite sum S = a_1 / (1 - r). Sigma notation writes a series compactly: SUM from k = 1 to n of f(k).
What you'll learn
- Compute the sum of a finite arithmetic series using S_n = (n/2)(a_1 + a_n)
- Compute the sum of a finite geometric series using S_n = a_1 * (1 - r^n) / (1 - r)
- Compute the sum of an infinite geometric series with |r| < 1 using S = a_1 / (1 - r)
Worked example
Problem. Find the sum of the first 10 terms of 3, 7, 11, 15, ...
- It is arithmetic: a_1 = 3, d = 4, so a_10 = 3 + 9 * 4 = 39.
- S_10 = (10 / 2)(3 + 39) = 5 * 42.
- = 210.
Answer: 210
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Arithmetic 2, 4, 6, 8, ... Find the sum of the first 10 terms.
Show solution
- a_10 = 2 + 9*2 = 20.
- S_10 = (10/2)(2 + 20) = 5 * 22 = 110.
Answer: 110
2. Arithmetic 1, 3, 5, 7, ... Find the sum of the first 10 terms.
Show solution
- a_10 = 1 + 9*2 = 19.
- S_10 = (10/2)(1 + 19) = 100.
Answer: 100
3. Geometric 1, 2, 4, 8, ... Find the sum of the first 5 terms.
Show solution
- S_5 = 1 * (1 - 2^5)/(1 - 2) = (1 - 32)/(-1) = 31.
Answer: 31
4. Geometric 2, 6, 18, 54. Find S_4.
Show solution
- a_1 = 2, r = 3.
- S_4 = 2 * (1 - 81)/(1 - 3) = 2 * (-80)/(-2) = 80.
Answer: 80
5. Infinite geometric series 1 + 1/2 + 1/4 + ... Find the sum.
Show solution
- a_1 = 1, r = 1/2.
- S = 1 / (1 - 1/2) = 1 / (1/2) = 2.
Answer: 2
Practice this interactively with instant feedback and an AI tutor.
Practice Arithmetic and Geometric Series Take the free placement check