Geometric Sequence Calculator
Calculate geometric sequence instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Geometric Sequence Calculator
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Formula: a_n = a * r^(n-1) | S_n = a(1 - r^n)/(1 - r)
Worked example — 10th term = 1,536 | Sum of 10 terms = 3,069
Formula
a_n = a * r^(n-1) | S_n = a(1 - r^n)/(1 - r)
Where a is the first term, r is the common ratio, n is the term position. For infinite series with |r| < 1, the sum converges to S = a/(1-r).
Worked Examples
Example 1: Finding the 10th Term and Sum
Problem:Find the 10th term and sum of the first 10 terms of the geometric sequence with a = 3 and r = 2.
Solution:a_10 = 3 * 2^(10-1) = 3 * 2^9 = 3 * 512 = 1,536 S_10 = 3 * (1 - 2^10) / (1 - 2) S_10 = 3 * (1 - 1024) / (-1) S_10 = 3 * 1023 = 3,069 Sequence: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536
Result:10th term = 1,536 | Sum of 10 terms = 3,069
Example 2: Convergent Infinite Series
Problem:Find the sum of the infinite geometric series: 100 + 50 + 25 + 12.5 + ...
Solution:First term a = 100, common ratio r = 50/100 = 0.5 Since |r| = 0.5 < 1, the series converges. S = a / (1 - r) = 100 / (1 - 0.5) S = 100 / 0.5 = 200 Verification: S_10 = 100 * (1 - 0.5^10) / 0.5 = 199.8 (very close to 200)
Result:Infinite sum = 200 | The series converges since |r| = 0.5 < 1
Frequently Asked Questions
What is a geometric sequence and how is it defined?
A geometric sequence (also called a geometric progression) is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio r. The general form is a, ar, ar^2, ar^3, and so on. For example, the sequence 3, 6, 12, 24, 48 is geometric with first term a = 3 and common ratio r = 2. Each term is exactly twice the previous one. Geometric sequences can grow (when |r| > 1), shrink toward zero (when |r| < 1), alternate in sign (when r < 0), or remain constant (when r = 1). They appear naturally in finance, biology, physics, and computer science.
How do you find the nth term of a geometric sequence?
The nth term of a geometric sequence is given by the formula a_n = a * r^(n-1), where a is the first term, r is the common ratio, and n is the position number. For example, in the sequence with a = 5 and r = 2, the 8th term is 5 * 2^7 = 5 * 128 = 640. This formula works because reaching the nth term requires multiplying by r exactly (n-1) times starting from the first term. When calculating large term numbers, the result can grow extremely fast for |r| > 1 or shrink rapidly for |r| < 1. This exponential nature is what distinguishes geometric sequences from arithmetic sequences, where each term differs by a constant addition.
What is the formula for the sum of a geometric series?
The sum of the first n terms of a geometric series is S_n = a * (1 - r^n) / (1 - r) when r is not equal to 1, and S_n = a * n when r = 1. For an infinite geometric series with |r| < 1, the sum converges to S = a / (1 - r). For example, the infinite series 1 + 1/2 + 1/4 + 1/8 + ... has a = 1 and r = 1/2, giving S = 1/(1 - 0.5) = 2. The finite sum formula is derived by multiplying S_n by r and subtracting from S_n to eliminate most terms. The infinite sum exists only when |r| < 1 because the terms diminish to zero fast enough that the total remains bounded.
What determines whether a geometric series converges or diverges?
A geometric series converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1). When |r| < 1, each successive term becomes smaller and smaller, approaching zero, and the partial sums approach a finite limit of a/(1-r). When |r| >= 1, the terms do not approach zero, so the series diverges (the partial sums grow without bound or oscillate). When r = -1, the series oscillates between two values and does not converge. When |r| = 1 and r is not -1, the series grows linearly. This convergence criterion is one of the simplest and most important tests in series analysis and provides a foundation for understanding more complex convergence tests.
How do you find the common ratio from two consecutive terms?
The common ratio r is found by dividing any term by its preceding term: r = a_(n+1) / a_n. For example, if consecutive terms are 12 and 36, then r = 36/12 = 3. If you have non-consecutive terms, say the 3rd term is 18 and the 7th term is 288, use r^(7-3) = 288/18 = 16, so r^4 = 16, giving r = 2 (or r = -2 if alternating signs are possible). You can also find the common ratio from the first term and the nth term: r = (a_n / a)^(1/(n-1)). Verifying the common ratio by checking multiple consecutive pairs helps confirm that the sequence is truly geometric and not just approximately geometric.
What are some real-world applications of geometric sequences?
Geometric sequences and series have extensive real-world applications across many fields. In finance, compound interest follows a geometric pattern where each period multiplies the balance by (1+r). Depreciation of assets at a fixed percentage rate creates a geometric decay sequence. In biology, population growth under ideal conditions follows geometric progression. In physics, the decay of radioactive materials produces a geometric sequence of remaining amounts. Musical frequencies of notes in equal temperament form a geometric sequence with ratio 2^(1/12). Fractal geometry uses geometric series to calculate perimeters and areas. Computer science uses geometric series in algorithm analysis, particularly for divide-and-conquer algorithms.
What is the geometric mean and how does it relate to geometric sequences?
The geometric mean of n numbers is the nth root of their product. For a geometric sequence, the geometric mean of consecutive terms equals the term between them. Specifically, for three consecutive terms a, ar, ar^2, the geometric mean is (a * ar * ar^2)^(1/3) = ar, which is the middle term. The geometric mean of the entire sequence a, ar, ar^2, ..., ar^(n-1) is a * r^((n-1)/2), which equals the square root of the product of the first and last terms. The geometric mean is always less than or equal to the arithmetic mean (AM-GM inequality) for positive numbers. It is used in finance to calculate average returns over time and in statistics for data that spans several orders of magnitude.
How do alternating geometric sequences behave?
When the common ratio r is negative, the geometric sequence alternates in sign: positive, negative, positive, negative, and so on. For example, with a = 1 and r = -2, the sequence is 1, -2, 4, -8, 16, -32, and so on. If |r| > 1 with r < 0, the terms grow in absolute value while alternating signs, and the partial sums oscillate with increasing amplitude. If |r| < 1 with r < 0 (like r = -0.5), the terms shrink toward zero while alternating, and the infinite series converges. The sum is a/(1-r) = 1/(1-(-0.5)) = 1/1.5 = 2/3. Alternating geometric sequences are important in signal processing, control theory, and the study of oscillating systems.
How do you insert geometric means between two numbers?
To insert k geometric means between two numbers a and b, you create a geometric sequence of k+2 terms starting at a and ending at b. The common ratio is r = (b/a)^(1/(k+1)). For example, to insert 3 geometric means between 2 and 162: r = (162/2)^(1/4) = 81^(1/4) = 3. The complete sequence is 2, 6, 18, 54, 162. This technique is useful in interpolation, creating smooth transitions, and constructing musical scales. In engineering, geometric interpolation ensures constant percentage changes between steps. In audio engineering, logarithmic frequency bands are created by inserting geometric means between frequency limits.
What is the relationship between geometric and arithmetic sequences?
Geometric and arithmetic sequences are related through logarithms and exponentials. If a sequence is geometric with terms a, ar, ar^2, ..., then taking the logarithm of each term gives log(a), log(a)+log(r), log(a)+2log(r), ..., which is an arithmetic sequence with common difference log(r). Conversely, if you have an arithmetic sequence and exponentiate each term, you get a geometric sequence. This relationship is why plotting geometric growth on a logarithmic scale produces a straight line. Both sequence types can be unified under the concept of an arithmo-geometric sequence, which has both additive and multiplicative components. Understanding both types is essential for analyzing patterns in data and solving recurrence relations.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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