a_n=a_1r^{n-1}Variables
- aₙ: nth term
- a₁: first term
- r: common ratio
- n: term number
How to use this formula
Finds the nth term of a sequence with a constant ratio.
Important notes
- Each term is obtained by multiplying the previous term by r.
- A negative ratio causes alternating signs.
Quick example
For 2, 6, 18, … the 5th term is 162.
Applicability, worked calculation, and verification
Domain and applicability
Applies to sequences with a constant multiplicative ratio between consecutive nonzero terms and integer index n ≥ 1.
Units
- a₁ and aₙ use the same units
- r and n are dimensionless
Assumptions and domain checks
- Each term is obtained by multiplying the previous term by r.
- For the Geometric Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Geometric Sequence Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Geometric Sequence Formula, check zero, negative, and extreme input values before relying on the result.
- When using Geometric Sequence Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form a_n=a_1r^{n-1} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For a sequence with first term 2 and common ratio 3, the fifth term is a₅ = 2·3⁴ = 162.
- Identify a₁ = 2, r = 3, and n = 5.
- Compute the exponent n − 1 = 4.
- Evaluate the repeated factor: 3⁴ = 81.
- Multiply by the first term: 2 × 81 = 162.
Independent verification
The sequence 2, 6, 18, 54, 162 multiplies by 3 at every step and confirms the fifth term.
Common mistakes
- Before substituting values into Geometric Sequence Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Geometric Sequence Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Geometric Sequence Formula in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
a_n=a_1r^{n-1}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Geometric Sequence Formula.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Geometric Sequence Formula used for?
Finds the nth term of a sequence with a constant ratio.
Can I copy this formula as LaTeX?
Yes. Copy a_n=a_1r^{n-1} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.