GP Terms and Sums

A geometric progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, 162, … is a GP with first term 2 and common ratio 3. Geometric progressions appear everywhere in mathematics and the real world — from compound interest and population growth to the decay of radioactive substances and the vibration of musical strings. Unlike an arithmetic progression where terms differ by a constant amount, a GP grows (or shrinks) multiplicatively. If the common ratio r satisfies |r| > 1, the terms grow without bound; if 0 < |r| < 1, the terms shrink toward zero; if r is negative, the terms alternate in sign. Understanding how to find any term in a GP and how to add up a finite or infinite collection of those terms is a foundational skill in algebra and calculus. This article covers the general term formula for a GP, the formula for the sum of a finite GP (called a finite geometric series), and the powerful result for the sum of an infinite GP when |r| < 1. Mastering these formulas will equip you to solve problems in finance, physics, and discrete mathematics.

Key takeaways
  • Identifying a Geometric Progression
  • The General (nth) Term
  • Sum of a Finite Geometric Series
  • Sum of an Infinite Geometric Series
  • Geometric Mean

Identifying a Geometric Progression

A sequence a₁, a₂, a₃, … is geometric if the ratio between consecutive terms is constant: a₂/a₁ = a₃/a₂ = … = r. To check whether a sequence is a GP, divide any term by the one before it — if you always get the same value, it is geometric. For example, in 5, 15, 45, 135: 15/5 = 3, 45/15 = 3, 135/45 = 3. The common ratio r = 3.

The General (nth) Term

Starting from a first term a and multiplying by r repeatedly: a₁ = a, a₂ = ar, a₃ = ar², a₄ = ar³. The pattern shows that the nth term is aₙ = a · r^(n−1). The exponent is always one less than the position number n. For instance, the 7th term of the GP with a = 3 and r = 2 is a₇ = 3 · 2⁶ = 3 · 64 = 192.

Sum of a Finite Geometric Series

Let Sₙ = a + ar + ar² + … + ar^(n−1). Multiply both sides by r: rSₙ = ar + ar² + … + ar^n. Subtract the first equation from the second: rSₙ − Sₙ = ar^n − a, so Sₙ(r − 1) = a(r^n − 1). Dividing gives Sₙ = a(r^n − 1)/(r − 1) for r ≠ 1. An equivalent form useful when r < 1 is Sₙ = a(1 − r^n)/(1 − r). When r = 1 all terms equal a, so Sₙ = na.

Sum of an Infinite Geometric Series

If |r| < 1, then r^n → 0 as n → ∞. Taking the limit of Sₙ = a(1 − r^n)/(1 − r) as n → ∞ gives S∞ = a/(1 − r). This is only valid when |r| < 1; if |r| ≥ 1 the series diverges (the partial sums grow without bound). For example, the infinite series 1 + 1/2 + 1/4 + 1/8 + … has a = 1, r = 1/2, so S∞ = 1/(1 − 1/2) = 2.

Geometric Mean

The geometric mean of two positive numbers p and q is √(pq). In a GP, every middle term is the geometric mean of its two neighbours: aₙ = √(aₙ₋₁ · aₙ₊₁). This property is often used in exam problems asking you to insert geometric means between two given values or to find an unknown term when others are given.

The Key Equation

an=a rn−1,Sn=a(1−rn)1−r  (r≠1),S∞=a1−r  (∣r∣<1)a_n = a\,r^{n-1}, \qquad S_n = \frac{a(1-r^n)}{1-r}\;(r\neq 1), \qquad S_\infty = \frac{a}{1-r}\;(|r|<1)
  • a — First term (dimensionless (or same units as the sequence)): The starting value of the geometric progression.
  • r — Common ratio (dimensionless): The fixed multiplier between consecutive terms; r ≠ 0.
  • n — Term number / number of terms (positive integer): The position of the desired term, or the count of terms being summed.
  • a_n — nth term (same as a): The value of the term at position n in the sequence.
  • S_n — Partial sum (same as a): The total obtained by adding the first n terms of the GP.
  • S_\infty — Infinite sum (same as a): The limit of the partial sums as n → ∞, defined only when |r| < 1.

Worked Example

Problem. A GP has first term a = 4 and common ratio r = 3. (a) Find the 6th term. (b) Find the sum of the first 6 terms. (c) A different GP has first term 8 and common ratio 1/2. Find the sum to infinity.

a6=972,S6=1456,S∞=16a_6 = 972,\quad S_6 = 1456,\quad S_\infty = 16

Answer. 6th term = 972; S₆ = 1456; S∞ = 16

Figure
Diagram
Geometric Progression: a=2, r=2 (first 6 terms)naₙ248163264122438416532664Each bar is double the previous — exponential (×2) growth
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-300–-260 · Greek

Euclid discussed geometric progressions and their properties in Books VIII and IX of his monumental work Elements (c. 300 BCE). He proved several propositions about numbers in continued proportion — the ancient Greek term for geometric sequences — including results about their sums and the conditions under which an infinite sum could be meaningful. His rigorous axiomatic approach influenced every generation of mathematicians that followed. Euclid's Elements was used as a mathematics textbook in schools and universities for over 2000 years — making it one of the most enduringly influential textbooks in history.

  • Arithmetic Progression: Terms and Sums — Arithmetic progressions are the additive counterpart to geometric progressions; comparing the two structures deepens understanding of both.
  • Convergence of Infinite Series — The infinite GP sum is the simplest example of a convergent series; understanding convergence generalises the S∞ formula.
  • Compound Interest — Compound interest generates a geometric progression of account balances, making GP formulas directly applicable to financial calculations.
  • Sigma Notation — Geometric series are commonly expressed using sigma (Σ) notation; fluency with both representations is essential.