Geometric Sequence and Series Formulas
Use geometric sequence formulas to find the common ratio, nth term, recursive rule, finite sum, and convergent infinite sum.
A geometric sequence changes by a constant ratio. Its powers control both individual terms and the sums of finite or convergent infinite series.
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5 formulas, different jobs
A geometric sequence changes by a constant ratio. Its powers control both individual terms and the sums of finite or convergent infinite series.
Common Ratio
Use it to find the fixed multiplier between consecutive terms.
Explicit Formula
Use it to find any term directly from its position.
Recursive Formula
Use it to generate each term from the previous term.
Finite Geometric Series
Use it to add a fixed number of geometric terms.
Infinite Geometric Series
Use it only when the terms shrink and the infinite series converges.
From a ratio to finite and infinite sums
For the sequence 81, 27, 9, 3, 1, …, find the fifth term, the first five-term sum, and the infinite sum.
- 1 · Ratio
- 2 · Fifth term
- 3 · Finite sum
- 4 · Infinite sum
Common mistakes
- Subtracting terms to find a common difference instead of dividing to find the ratio.
- Using n instead of n − 1 in the explicit formula.
- Applying the infinite-sum formula when |r| is greater than or equal to 1.
See which formulas your problem uses
Enter a problem and get a step-by-step solution with the formulas and theorems identified.
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Arithmetic sequences
Use arithmetic sequence formulas to find the common difference, nth term, recursive rule, and sum of a finite arithmetic series.
Exponent rules
Learn the product, quotient, power, zero, negative, and fractional exponent rules with examples and one complete simplification.
Geometric sequences FAQ
What makes a sequence geometric?
The ratio between every pair of consecutive nonzero terms must be the same constant r.
When does an infinite geometric series have a sum?
It converges only when the absolute value of the common ratio is less than 1.