Geometric Progression Calculator
Calculate nth term, sum, and properties of geometric sequences and series. Enter values for instant results with step-by-step formulas.
Calculator
Adjust values & calculateSequence Terms
Formula
Where a_n = nth term, a = first term, r = common ratio, n = number of terms, S_n = sum of first n terms (for r not equal to 1), and S_inf = sum of infinite series (only when |r| < 1). The nth term formula multiplies the first term by the ratio raised to the (n-1) power.
Last reviewed: December 2025
Worked Examples
Example 1: Compound Investment Growth
Example 2: Bouncing Ball Distance
Background & Theory
The Geometric Progression Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Geometric Progression Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
a_n = a * r^(n-1) | S_n = a(r^n - 1)/(r - 1) | S_inf = a/(1-r)
Where a_n = nth term, a = first term, r = common ratio, n = number of terms, S_n = sum of first n terms (for r not equal to 1), and S_inf = sum of infinite series (only when |r| < 1). The nth term formula multiplies the first term by the ratio raised to the (n-1) power.
Worked Examples
Example 1: Compound Investment Growth
Problem: An investment of $5,000 grows at 8% annually. Find the value after 10 years and the total growth factor. Also calculate the sum of all year-end values.
Solution: First term a = 5000, Common ratio r = 1.08, n = 10\n10th term (value after 10 years): a_10 = 5000 x 1.08^9 = 5000 x 1.999 = $9,995.02\nNote: after 10 FULL years, value = 5000 x 1.08^10 = $10,794.62\nSum of year-end values (years 1-10):\nS_10 = 5000(1.08^10 - 1)/(1.08 - 1)\n= 5000(2.1589 - 1)/0.08\n= 5000 x 14.487 = $72,432.81\nTotal growth factor: 1.08^10 = 2.159
Result: Year 10 value: $10,794.62 | Sum of all values: $72,432.81 | Growth: 2.16x
Example 2: Bouncing Ball Distance
Problem: A ball is dropped from 10 meters and each bounce reaches 60% of the previous height. Find the total distance traveled before it stops.
Solution: The ball falls 10m, bounces up 6m, falls 6m, bounces up 3.6m, etc.\nDownward distances: 10, 6, 3.6, 2.16, ... (GP with a=10, r=0.6)\nUpward distances: 6, 3.6, 2.16, ... (GP with a=6, r=0.6)\nTotal down = 10/(1-0.6) = 10/0.4 = 25 meters\nTotal up = 6/(1-0.6) = 6/0.4 = 15 meters\nTotal distance = 25 + 15 = 40 meters\nAlternatively: Total = 10 + 2(6)/(1-0.6) = 10 + 30 = 40 meters
Result: Total distance: 40 meters | Infinite bounces converge to finite distance
Frequently Asked Questions
What is a geometric progression and what defines it?
A geometric progression (GP), also called a geometric sequence, is an ordered list of numbers where each term after the first is obtained by multiplying the previous term by a fixed nonzero number called the common ratio, denoted by r. If the first term is a and the common ratio is r, the sequence is a, ar, ar^2, ar^3, and so on. For example, 3, 6, 12, 24, 48 is a geometric progression with first term 3 and common ratio 2. Unlike arithmetic progressions where terms grow linearly, geometric progressions exhibit exponential growth or decay. When the common ratio is between -1 and 1 (exclusive), the terms decrease in absolute value toward zero.
How do you find the nth term of a geometric progression?
The nth term of a geometric progression is given by the formula a_n = a times r^(n-1), where a is the first term, r is the common ratio, and n is the position number. This formula works because each step multiplies by r, so reaching the nth term requires (n-1) multiplications by r starting from a. For example, in the GP 2, 6, 18, 54, ..., the 8th term is 2 times 3^7 = 2 times 2187 = 4374. Geometric progressions grow or shrink much faster than arithmetic ones: a GP with ratio 2 doubles every step, reaching over a million by the 20th term from just 1. This exponential behavior makes GPs essential for modeling compound growth phenomena.
What is the formula for the sum of a finite geometric series?
The sum of the first n terms of a geometric series is S_n = a(r^n - 1)/(r - 1) when r is not equal to 1, or S_n = na when r equals 1. This formula is derived by multiplying the series by r, subtracting the original from the result, and solving for S. For example, the sum of the first 6 terms of the GP 3, 6, 12, 24, 48, 96 is S_6 = 3(2^6 - 1)/(2 - 1) = 3(64 - 1)/1 = 3 times 63 = 189. An equivalent form is S_n = a(1 - r^n)/(1 - r), which is more convenient when |r| < 1. The formula works for any real common ratio except r = 1, and the choice of which form to use depends on whether r is greater or less than 1.
When does an infinite geometric series converge and what is its sum?
An infinite geometric series converges (has a finite sum) if and only if the absolute value of the common ratio is less than 1, that is |r| < 1. When this condition is met, the infinite sum is S = a / (1 - r). As n approaches infinity, r^n approaches zero when |r| < 1, causing the partial sum formula S_n = a(1 - r^n)/(1 - r) to approach a/(1 - r). For example, the infinite series 1 + 1/2 + 1/4 + 1/8 + ... has sum = 1/(1 - 0.5) = 2. When |r| >= 1, the terms do not decrease toward zero, so the series diverges. This convergence property has profound applications in mathematics, physics, and finance, from calculating present values of perpetuities to analyzing bouncing ball distances.
What is the geometric mean and how does it relate to geometric progressions?
The geometric mean of two positive numbers p and q is sqrt(p times q), and it is the value that, when placed between p and q, forms a three-term geometric progression with equal ratios. More generally, inserting k geometric means between two positive numbers a and b creates a GP of k+2 terms, where the common ratio r = (b/a)^(1/(k+1)). For example, inserting 2 geometric means between 2 and 54 gives r = (54/2)^(1/3) = 27^(1/3) = 3, producing the sequence 2, 6, 18, 54. In a GP, every term (except the first and last) equals the geometric mean of its two neighbors. The geometric mean is always less than or equal to the arithmetic mean for positive numbers, with equality only when both numbers are the same.
What are common real-world applications of geometric progressions?
Geometric progressions model any phenomenon involving constant percentage change over equal intervals. Compound interest is the most classic example: an investment growing at 5% annually follows a GP with ratio 1.05. Population growth and decay, including bacterial growth and radioactive decay, follow geometric patterns. In music, the frequencies of notes in an equal-tempered scale form a GP with ratio 2^(1/12) because each semitone multiplies the frequency by this constant. Computer science uses GPs in algorithm analysis, where doubling input size has multiplicative effects on runtime. Signal attenuation in telecommunications, drug concentration decay in pharmacology, and mortgage amortization all involve geometric progressions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy