Geometric Sequences and Recursive Rules

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

Learn with original explanations and worked examples. OpenStax Elementary Algebra 2e was consulted for topic coverage.

Study every current unit through original concept lessons, formula derivations, experiments, quizzes, and practice sets.

Algebra 1 · Unit 10 · Topic 10.2

Geometric Sequences and Recursive Rules

A geometric sequence multiplies by a fixed ratio r each step.

Definition and explanation

A geometric sequence multiplies by a fixed ratio r each step. Its explicit rule is aₙ = a₁rⁿ⁻¹ and its recursive rule is aₙ = raₙ₋₁ for n ≥ 2, together with the initial value a₁.

Worked example

For 81, 27, 9, 3, the ratio is 1/3. The rule aₙ = 81(1/3)ⁿ⁻¹ gives a₅ = 81/81 = 1. Recursively, multiplying the fourth term 3 by 1/3 gives the same result.

Common mistake

A recursive rule without its starting value does not determine a unique sequence.

Key takeaway

A geometric sequence multiplies by a fixed ratio r each step.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.