Geometric Sequences
A free Algebra I lesson from the “Exponential Functions and Sequences” unit, with a worked example and practice problems including step-by-step solutions.
A geometric sequence multiplies by the same number r (the common ratio) to get from one term to the next. The explicit formula a_n = a_1 * r^(n - 1) finds a term directly from its position n. A recursive rule gives the first term a_1, then a_n = r * a_(n - 1) for whole-number positions n >= 2. The symbol a_(n - 1) means the previous term; multiply that term by r, rather than multiplying the position n. Always include the starting value. For positive terms, r > 1 gives growth and 0 < r < 1 gives decay; a negative ratio alternates the signs of nonzero terms.
What you'll learn
- Identify a geometric sequence by its constant ratio
- Use the explicit formula a_n = a_1 * r^(n - 1)
- Write a recursive rule with a first term and a_n = r * a_(n - 1) for n >= 2
- Find a specific term given the first term and the common ratio
Worked example
Problem. For the geometric sequence 2, 6, 18, 54, ..., write a recursive rule, use it to find a_3, and use an explicit formula to find a_5.
- The first term is 2 and the common ratio is 6 / 2 = 3.
- Recursive rule: a_1 = 2; a_n = 3 * a_(n - 1) for n >= 2.
- Build from the first term: a_2 = 3 * 2 = 6, then a_3 = 3 * 6 = 18.
- The equivalent explicit rule is a_n = 2 * 3^(n - 1). Thus a_5 = 2 * 3^4 = 162.
Answer: a_1 = 2; a_n = 3 * a_(n - 1) for n >= 2; a_3 = 18; a_5 = 162
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. For 1, 2, 4, 8, ..., find r.
Show solution
- 2 / 1 = 2.
Answer: 2
2. For 1, 2, 4, 8, ..., find a_6.
Show solution
- a_6 = 1 * 2^5 = 32.
Answer: 32
3. For 81, 27, 9, 3, ..., find r as a fraction.
Show solution
- 27 / 81 = 1/3.
Answer: 1/3
4. For 5, 10, 20, 40, ..., find a_5.
Show solution
- r = 2; a_5 = 5 * 2^4 = 80.
Answer: 80
5. For 100, 50, 25, ..., find r as a fraction.
Show solution
- 50 / 100 = 1/2.
Answer: 1/2
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