Arithmetic 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.
An arithmetic sequence adds the same number d (the common difference) to get from one term to the next. The explicit formula a_n = a_1 + (n - 1) * d finds a term directly from its position n. A recursive rule instead uses the previous term: state the first term a_1, then a_n = a_(n - 1) + d for whole-number positions n >= 2. Here a_(n - 1) means the term just before a_n. Both the starting value and the recurrence are needed to specify the sequence. A negative d subtracts the same amount at each step.
What you'll learn
- Identify an arithmetic sequence by its constant difference
- Use the explicit formula a_n = a_1 + (n - 1) * d
- Write a recursive rule with a first term and a_n = a_(n - 1) + d for n >= 2
- Find a specific term given the first term and the common difference
Worked example
Problem. For the arithmetic sequence 3, 7, 11, 15, ..., write a recursive rule, use it to find a_3, and use an explicit formula to find a_10.
- The first term is 3 and the common difference is 7 - 3 = 4.
- Recursive rule: a_1 = 3; a_n = a_(n - 1) + 4 for n >= 2.
- Build from the first term: a_2 = 3 + 4 = 7, then a_3 = 7 + 4 = 11.
- The equivalent explicit rule is a_n = 3 + (n - 1) * 4. Jump to a_10 = 3 + 9 * 4 = 39.
Answer: a_1 = 3; a_n = a_(n - 1) + 4 for n >= 2; a_3 = 11; a_10 = 39
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 2, 5, 8, 11, ..., find d.
Show solution
- 5 - 2 = 3.
Answer: 3
2. For 2, 5, 8, 11, ..., find a_10.
Show solution
- a_10 = 2 + 9 * 3 = 29.
Answer: 29
3. For 100, 95, 90, ..., find d.
Show solution
- 95 - 100 = -5.
Answer: -5
4. For 100, 95, 90, ..., find a_5.
Show solution
- a_5 = 100 + 4 * (-5) = 80.
Answer: 80
5. For 4, 9, 14, 19, ..., find a_6.
Show solution
- a_6 = 4 + 5 * 5 = 29.
Answer: 29
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