Geometric Sequence
Find the nth term, sum of geometric series, and full sequence of any geometric progression. Free online geometric sequence calculator with step-by-step results.
About This Calculator
A geometric sequence (also called a geometric progression) is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed constant known as the common ratio, denoted as r. For example, the sequence 2, 6, 18, 54, 162 is geometric because each term is multiplied by 3 to get the next term.
Our geometric sequence calculator computes three key results instantly. The nth term (aₙ) is calculated using the formula aₙ = a1 x rⁿ⁻¹. The sum of the first n terms (Sₙ) is given by Sₙ = a1(1-rⁿ)/(1-r) when r ≠ 1, and Sₙ = a1 x n when r = 1. The calculator also displays the full sequence as a comma-separated list so you can inspect every term.
This calculator handles any real-number inputs, including positive, negative, fractional, and decimal ratios. For the common ratio r = 0, all terms after the first become zero. For r = 1, all terms equal the first term. For |r| < 1, the terms decrease toward zero. For |r| > 1, the terms grow exponentially. The calculator supports large values of n, limited only by your browser's memory.
Regional Notes
Geometric sequences are taught worldwide in secondary mathematics. In India, they are part of CBSE Class 11 mathematics under sequences and series. In the US, geometric sequences appear in Algebra II and Precalculus. In the UK, they are covered in GCSE and A-Level mathematics. The formulas are standard across all educational systems.
Frequently Asked Questions
What is a geometric sequence?
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio (r). For example, 3, 6, 12, 24, 48 is a geometric sequence with first term 3 and common ratio 2.
How do you find the nth term of a geometric sequence?
The nth term of a geometric sequence is given by the formula aₙ = a1 x rⁿ⁻¹, where a1 is the first term, r is the common ratio, and n is the term position. Enter these values into the calculator to get the nth term instantly.
What is the sum of a geometric series formula?
The sum of the first n terms of a geometric series is Sₙ = a1(1-rⁿ)/(1-r) when r ≠ 1. If r = 1, the sum is simply a1 x n. For infinite geometric series where |r| < 1, the sum converges to a1/(1-r).
Can the common ratio be a fraction or decimal?
Yes, the common ratio can be any real number. A ratio between 0 and 1 produces a decreasing sequence (e.g., 16, 8, 4, 2, 1 with r = 0.5). A negative ratio creates an alternating sequence where terms switch signs. A ratio of 1 gives a constant sequence.
What is the difference between arithmetic and geometric sequences?
In an arithmetic sequence, terms increase by a constant difference (addition). In a geometric sequence, terms increase by a constant ratio (multiplication). Arithmetic sequences grow linearly, while geometric sequences can grow exponentially or decay depending on the common ratio.
What real-world applications use geometric sequences?
Geometric sequences model exponential growth and decay: compound interest on investments, population growth, radioactive decay, depreciation of assets, and viral spread of information. They are also used in calculating mortgage payments and understanding geometric series in finance.
How do you calculate the sum of an infinite geometric series?
An infinite geometric series converges to a finite sum only when |r| < 1. The sum is S = a1/(1-r). For example, the series 1 + 1/2 + 1/4 + 1/8 + ... has sum S = 1/(1-0.5) = 2. If |r| >= 1, the infinite series diverges and has no finite sum.