In short

A geometric progression (GP) is a sequence in which every term after the first is obtained by multiplying the previous term by a fixed number called the common ratio r. If the first term is a, the n-th term is a \cdot r^{\,n-1}. While an arithmetic progression grows by adding, a GP grows by multiplying — and that single change turns steady staircase growth into explosive (or collapsing) exponential behaviour.

You deposit ₹10,000 in a savings account that pays 10% annual interest. After one year you have ₹11,000. After two years you do not have ₹12,000 — you have ₹12,100, because the interest in the second year is computed on ₹11,000, not on the original ₹10,000. After three years, ₹13,310. After four years, ₹14,641.

Line the balances up: 10000,\; 11000,\; 12100,\; 13310,\; 14641. Each term is 1.1 times the one before. The differences between consecutive terms are not constant — the jumps get bigger as the balance grows. This is not an arithmetic progression. It is a geometric progression, and the fixed multiplier 1.1 is its common ratio.

Understanding GPs means understanding how anything that grows (or shrinks) by a fixed percentage at each step behaves over time — compound interest, population growth, radioactive decay, the number of bacteria doubling every hour.

Definition and common ratio

A geometric progression is a sequence of the form

a, \; ar, \; ar^2, \; ar^3, \; \dots

where a is the first term and r is the common ratio — the factor you multiply by to get from one term to the next.

First five terms of a generic geometric progressionFive boxes in a row labelled a, ar, ar squared, ar cubed, and ar to the fourth. Arrows between consecutive boxes are each labelled times r, showing the constant multiplication at each step. a ar ar² ar³ ar⁴ ×r ×r ×r ×r each term = previous term × r
A GP is built by repeated multiplication. Starting from $a$, each step multiplies by the same ratio $r$. The $n$-th term is $a \cdot r^{\,n-1}$ — exponents count the number of multiplications you have done since the start.

The common ratio is found by dividing any term by the one before it:

r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \cdots

If this ratio is the same for every consecutive pair, the sequence is a GP. If it varies, it is not.

Some quick examples:

Sequence a r Pattern
2, 6, 18, 54, \dots 2 3 each term tripled
100, 50, 25, 12.5, \dots 100 0.5 each term halved
-3, 6, -12, 24, \dots -3 -2 sign alternates
5, 5, 5, 5, \dots 5 1 constant sequence

The common ratio r can be any non-zero real number. When r > 1, the terms grow without bound. When 0 < r < 1, the terms shrink toward zero. When r < 0, the terms alternate in sign. When r = 1, the GP is a constant sequence — technically still geometric, but not very interesting.

The one value r cannot take is 0. If r = 0, every term after the first would be zero, and you would lose the ability to recover r by dividing consecutive terms.

Geometric Progression

A sequence a_1, a_2, a_3, \dots is a geometric progression with first term a = a_1 and common ratio r \neq 0 if

a_{n+1} = r \cdot a_n \quad \text{for all } n \geq 1

Equivalently, the ratio \dfrac{a_{n+1}}{a_n} is the same constant r for every n.

The n-th term formula

Start from the definition and unroll:

a_1 = a
a_2 = a \cdot r
a_3 = a \cdot r \cdot r = a \cdot r^2
a_4 = a \cdot r^3

The pattern is immediate: to get from the first term to the n-th, you multiply by r exactly n - 1 times.

\boxed{a_n = a \cdot r^{\,n-1}}

The exponent is n - 1, not n, because the first term has had zero multiplications. This is worth memorising cold — it is the GP counterpart of the AP formula a_n = a + (n - 1)d, with multiplication replacing addition and exponentiation replacing repeated addition.

For the savings account example: a = 10000, r = 1.1, so the balance after n years (which is the (n + 1)-th term) is 10000 \times 1.1^{\,n}. After 4 years: 10000 \times 1.1^4 = 10000 \times 1.4641 = 14641.

AP versus GP growth shown as bar chartsTwo side-by-side bar charts, each with six bars representing terms 1 through 6. The left chart shows an AP with first term 2 and common difference 3, producing bars of heights 2, 5, 8, 11, 14, 17 — evenly increasing. The right chart shows a GP with first term 2 and common ratio 2, producing bars of heights 2, 4, 8, 16, 32, 64 — rapidly accelerating. The contrast shows how multiplicative growth outpaces additive growth. AP: a=2, d=3 2 5 8 11 14 17 GP: a=2, r=2 2 4 8 16 32 64
Six terms of an AP ($a = 2, d = 3$) versus six terms of a GP ($a = 2, r = 2$). The AP bars grow by the same height at every step — a straight staircase. The GP bars double each time, and by the sixth term the GP has overtaken the AP by a factor of nearly four. This is the visual signature of exponential growth.

Properties of GP

Property 1: Constant ratio

This is the defining property, but it is worth restating for problem-solving. If three numbers a, b, c are in GP, then

\frac{b}{a} = \frac{c}{b} \qquad \Longrightarrow \qquad b^2 = ac

The middle term squared equals the product of the outer two. This is the GP analogue of the AP property "twice the middle term equals the sum of the outer two." The condition b^2 = ac is the quickest test for checking whether three given numbers form a GP.

Property 2: Products and quotients of GP terms

If a_1, a_2, a_3, \dots is a GP, then:

These follow directly from the laws of exponents: a_n = a \cdot r^{n-1}, so a_n^m = a^m \cdot (r^m)^{n-1}, which is a GP with first term a^m and ratio r^m.

Property 3: Terms equidistant from the ends

In a finite GP a_1, a_2, \dots, a_n, the product of terms equidistant from the two ends is constant:

a_1 \cdot a_n = a_2 \cdot a_{n-1} = a_3 \cdot a_{n-2} = \cdots

Each of these products equals a^2 \cdot r^{\,n-1}. The proof: a_k \cdot a_{n+1-k} = (a \cdot r^{k-1})(a \cdot r^{n-k}) = a^2 \cdot r^{n-1}, which is the same for every k.

This is the multiplicative cousin of the AP property where the sum of equidistant terms is constant.

Property 4: Three consecutive terms

A convenient trick for setting up GP problems. If three terms of a GP are needed, write them as \dfrac{a}{r},\; a,\; ar. Their product is a^3 — independent of r — so if the product is given, you can find a immediately.

Similarly, for four terms, use \dfrac{a}{r^3},\; \dfrac{a}{r},\; ar,\; ar^3 (ratio r^2 between consecutive terms). The product of all four is a^4, again independent of r.

Three GP terms written as a over r, a, ar with product a cubedThree boxes labelled a over r, a, and ar arranged in a row. Curved arrows between them are labelled times r. Below the three boxes, a brace groups them and shows that their product equals a cubed. a / r a a r ×r ×r product = a³
Writing three GP terms as $\dfrac{a}{r}, a, ar$ centres them on $a$. The product is $\dfrac{a}{r} \cdot a \cdot ar = a^3$, with the $r$'s cancelling. This makes $a$ easy to find whenever the product of three terms in GP is given.

Property 5: Geometric means

If a, b, c are in GP, then b is the geometric mean of a and c: b = \sqrt{ac} (taking the positive root when a and c are positive). This is the GP counterpart of the AM being the middle term of a three-term AP. The article on Geometric Mean develops this fully.

Different shapes of a GP

The behaviour of a GP depends entirely on the value of r. The figure below shows five GPs, all starting at a = 16, with different common ratios. The vertical axis is the term value; the horizontal axis is the term number.

Five GPs with different common ratios starting from the same first termA coordinate plane with term number on the horizontal axis from 1 to 7 and term value on the vertical axis from negative 40 to 80. Five curves are plotted: r equals 1.5, which rises steeply; r equals 1, which is flat at 16; r equals 0.5, which decays toward zero; r equals negative 0.5, which oscillates with decreasing amplitude; and r equals negative 1, which alternates between 16 and negative 16. 1 2 3 4 5 6 7 0 16 32 48 −16 r = 1.5 r = 1 r = 0.5 r = −0.5 r = −1
Five GPs all starting at $a = 16$. When $r > 1$ (red), the terms explode upward. When $r = 1$, the GP is flat. When $0 < r < 1$, the terms decay toward zero. When $-1 < r < 0$, the terms oscillate with shrinking amplitude. When $r = -1$, the terms alternate between $16$ and $-16$ forever.

The key observation: when |r| < 1, the terms get smaller and smaller, approaching zero. When |r| > 1, the terms grow without bound. When |r| = 1, the terms stay the same size (either constant or alternating between a and -a). This trichotomy is the foundation of the infinite GP sum formula in Sum of Geometric Progression.

Two worked examples

Example 1: The 3rd term of a GP is 12 and the 6th term is 96. Find the first term and the common ratio.

This problem gives you two terms and asks you to recover the GP's parameters.

Step 1. Write the two given terms using the n-th term formula.

a_3 = a \cdot r^2 = 12
a_6 = a \cdot r^5 = 96

Why: the n-th term of a GP is a \cdot r^{n-1}, so the 3rd term has exponent 2 and the 6th has exponent 5.

Step 2. Divide the second equation by the first to eliminate a.

\frac{a \cdot r^5}{a \cdot r^2} = \frac{96}{12}
r^3 = 8

Why: dividing two terms of a GP cancels the first term a, leaving a pure power of r. The exponent in r^3 is the difference 5 - 2 = 3.

Step 3. Solve for r.

r = \sqrt[3]{8} = 2

Why: the cube root of 8 is 2, using the laws from Exponents and Powers.

Step 4. Substitute back to find a.

a \cdot 2^2 = 12 \implies 4a = 12 \implies a = 3

Result: The first term is a = 3 and the common ratio is r = 2. The GP is 3, 6, 12, 24, 48, 96, \dots

GP with first term 3 and ratio 2 showing the 3rd and 6th terms highlightedA bar chart with six bars of heights proportional to 3, 6, 12, 24, 48, and 96. The 3rd bar at height 12 and the 6th bar at height 96 are shaded red to indicate the given information. The other bars are shaded in dark grey as computed values. 3 6 12 24 48 96 3rd term 6th term each bar = 2 × previous
The GP $3, 6, 12, 24, 48, 96$ with the two given terms (3rd and 6th) highlighted in red. Each bar is exactly twice the previous one, confirming $r = 2$. The doubling is hard to see for the first few terms but unmistakable by the 6th.

The technique of dividing two known terms to isolate r works whenever you are given any two terms of a GP. If a_m and a_n are given (with m < n), then r^{n - m} = \dfrac{a_n}{a_m}, and r follows by taking the (n - m)-th root.

Example 2: Three numbers in GP have a product of 216 and a sum of 26. Find them.

This calls for the centred-form trick.

Step 1. Write the three terms as \dfrac{a}{r},\; a,\; ar.

Why: this form guarantees the three terms are in GP with common ratio r, and it makes the product easy to handle.

Step 2. Use the product.

\frac{a}{r} \cdot a \cdot ar = a^3 = 216
a = \sqrt[3]{216} = 6

Why: in the centred form, the r's cancel from the product, leaving only a^3.

Step 3. Use the sum with a = 6.

\frac{6}{r} + 6 + 6r = 26
\frac{6}{r} + 6r = 20

Multiply through by r:

6 + 6r^2 = 20r
6r^2 - 20r + 6 = 0
3r^2 - 10r + 3 = 0

Why: multiplying by r clears the fraction, giving a standard quadratic in r.

Step 4. Solve the quadratic.

r = \frac{10 \pm \sqrt{100 - 36}}{6} = \frac{10 \pm 8}{6}
r = 3 \quad \text{or} \quad r = \frac{1}{3}

Why: both roots give valid GPs — the two values of r produce the same three numbers in reverse order.

If r = 3: the terms are \dfrac{6}{3} = 2, 6, 6 \times 3 = 18. Check: 2 + 6 + 18 = 26 and 2 \times 6 \times 18 = 216.

If r = \dfrac{1}{3}: the terms are 18, 6, 2 — the same three numbers reversed.

Result: The three numbers are 2, 6, 18.

Three numbers 2, 6, 18 in GP with ratio 3 on a number lineA number line from 0 to 20 with three red points at 2, 6, and 18. Curved arrows above the line connect 2 to 6 and 6 to 18, each labelled times 3. Below the line, the product 216 and sum 26 are shown as verification. 0 10 20 2 6 18 ×3 ×3 product = 2 × 6 × 18 = 216 sum = 2 + 6 + 18 = 26
The three GP terms $2, 6, 18$ on the number line. The ratio between consecutive terms is $3$. The gaps between the points are not equal (the jump from $6$ to $18$ is much larger than from $2$ to $6$) — that is the signature of a GP, where the *ratio* is constant, not the *difference*.

Common confusions

Going deeper

If you came here to learn the definition, the n-th term formula, and the basic properties of GPs, you have them. The rest of this section connects GPs to logarithms and to the history of an idea that transformed computation.

GPs and logarithms

Take the GP a, ar, ar^2, ar^3, \dots and apply a logarithm to every term:

\log a, \quad \log a + \log r, \quad \log a + 2\log r, \quad \log a + 3\log r, \quad \dots

The result is an arithmetic progression with first term \log a and common difference \log r. Logarithms turn GPs into APs and multiplication into addition. This is the fundamental link between the two progressions, and it is the reason logarithms were invented in the first place — before electronic calculators, turning a hard multiplication into an easy addition was the most valuable trick in computational mathematics. The article on Exponents and Powers lays the groundwork for this connection.

Infinite GPs and convergence

When |r| < 1, the terms ar^n shrink toward zero as n grows. The GP 10, 5, 2.5, 1.25, \dots (with r = 0.5) never actually reaches zero, but the terms get as close to zero as you like. You can ask: what happens if you add up all the terms, forever?

The answer is that the sum converges to a finite number. For the GP above, 10 + 5 + 2.5 + 1.25 + \cdots = 20. The formula and its derivation are the subject of Sum of Geometric Progression, which is the next article in this sequence.

When |r| \geq 1, the terms do not shrink, and the sum does not converge. The distinction between |r| < 1 (convergent) and |r| \geq 1 (divergent) is one of the first encounters you will have with the concept of convergence — an idea that runs through all of calculus and analysis.

Compound growth as a GP

The savings-account example from the opening is the template for every compound growth problem. If a quantity starts at P and grows by a factor of r at each step:

This is why GPs matter beyond textbook exercises. Any process where each step multiplies by a constant — population growth, inflation, radioactive decay, the spread of a rumour — follows a GP. The explosive growth you saw in the bar chart is the same explosive growth that makes compound interest powerful and pandemics dangerous.

Where this leads next

Geometric progressions are the gateway to every topic involving exponential growth and multiplicative structure.