What is a geometric sequence?

A geometric sequence is a sequence in which consecutive terms have a constant ratio called the common ratio, denoted r. Starting from a nonzero first term a_1, each subsequent term is obtained by multiplying the previous one by r, producing the pattern a_1, a_1 r, a_1 r squared, and so on.

What is the formula for the nth term of a geometric sequence?

The general term of a geometric sequence is a_n = a_1 times r raised to the power n minus 1, where a_1 is the first term, r is the common ratio, and n is the position. This explicit formula gives any term directly without computing the ones before it.

What is the sum of a geometric series?

The sum of the first n terms of a geometric sequence is S_n = a_1 times (1 minus r to the power n) divided by (1 minus r), valid when r is not equal to 1. When r equals 1, every term is a_1 and the sum is simply n times a_1. The formula is derived using a telescoping subtraction trick.

When does an infinite geometric series converge?

An infinite geometric series converges when the absolute value of the common ratio r is less than 1. In that case, the sum equals a_1 divided by (1 minus r). When the absolute value of r is greater than or equal to 1, the terms do not diminish and the series diverges with no finite sum.

What is the geometric mean in a geometric sequence?

The geometric mean of two positive numbers a and b is the square root of their product. In a geometric sequence with positive terms, every interior term equals the geometric mean of its two neighbors. The AM-GM inequality states that the geometric mean never exceeds the arithmetic mean, with equality only when all values are identical.

Geometric Sequences

Definition

Identifying Geometric Sequences

Finding Terms

Recursive Formula

Finite Geometric Series

Infinite Geometric Series

Geometric Mean

Constant Ratio

A geometric sequence multiplies each term by the same fixed factor to produce the next. That factor — the common ratio — controls whether the sequence grows, decays, oscillates, or converges. Geometric sequences appear wherever quantities scale by a constant proportion, and their series formulas include one of the few cases where an infinite sum has a finite closed form.

Definition

A geometric sequence is a sequence in which consecutive terms have a constant ratio

r

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

Starting from a first term

a_1 \neq 0

, the sequence unfolds as

a_1, a_1 r, a_1 r^2, a_1 r^3, \ldots

. The general term is:

a_n = a_1 \cdot r^{n-1}

The ratio

r

determines the sequence's character. When

r > 1

, the terms grow without bound:

2, 6, 18, 54, \ldots

with

r = 3

. When

0 < r < 1

, the terms decay toward zero:

81, 27, 9, 3, 1, \ldots

with

r = \frac{1}{3}

. When

r < 0

, the terms alternate in sign:

4, -8, 16, -32, \ldots

with

r = -2

.

The case

r = 1

produces a constant sequence — every term equals

a_1

. The case

r = 0

is excluded because every term after the first would be zero, and the ratio

\frac{a_{n+1}}{a_n}

would be undefined.

Where an arithmetic sequence is linear — its terms lie on a straight line when plotted against their indices — a geometric sequence is exponential. The general term

a_1 \cdot r^{n-1}

is an exponential function of

n

Identifying Geometric Sequences

To test whether a sequence is geometric, compute the ratio of consecutive terms. If every ratio equals the same value, the sequence is geometric and that value is

r

.

The sequence

5, 15, 45, 135

has ratios

3, 3, 3

. All equal, so it is geometric with

r = 3

.

The sequence

12, 6, 3, 1.5

has ratios

0.5, 0.5, 0.5

. Geometric with

r = \frac{1}{2}

.

The sequence

1, 2, 4, 7

has ratios

2, 2, 1.75

. The final ratio differs, so the sequence is not geometric.

A constant sequence like

5, 5, 5, 5

is both arithmetic (with

d = 0

) and geometric (with

r = 1

). This is the only type of sequence that belongs to both families.

As with arithmetic sequences, a finite sample cannot confirm a geometric pattern with certainty. The sequence

2, 4, 8

could continue as

16, 32, \ldots

(geometric with

r = 2

) or as

14, 22, \ldots

(not geometric). The defining rule, not the sample, is what settles the classification.

Finding Terms

The explicit formula

a_n = a_1 \cdot r^{n-1}

contains four quantities. Given any three, the fourth can be found.

Finding a specific term: if

a_1 = 3

and

r = 2

, then

a_{10} = 3 \cdot 2^9 = 1536

.

Finding the common ratio: if

a_1 = 5

and

a_4 = 40

, then

40 = 5 \cdot r^3

, so

r^3 = 8

and

r = 2

.

Finding the first term: if

a_6 = 96

and

r = 2

, then

96 = a_1 \cdot 2^5

, so

a_1 = 3

.

Finding the position of a term: if

a_1 = 4

r = 3

, and some term equals

2916

, then

2916 = 4 \cdot 3^{n-1}

, so

3^{n-1} = 729 = 3^6

and

n = 7

. When the arithmetic does not produce a clean power, solving for

n

requires logarithms:

n - 1 = \frac{\ln(a_n / a_1)}{\ln r}

.

When two non-consecutive terms are given — say

a_2 = 6

and

a_5 = 162

— write both in terms of

a_1

and

r

a_1 r = 6

and

a_1 r^4 = 162

. Dividing gives

r^3 = 27

, so

r = 3

, and back-substituting gives

a_1 = 2

Recursive Formula

The recursive definition of a geometric sequence specifies a starting value and a multiplicative rule:

a_1 = c, \quad a_n = r \cdot a_{n-1}

Each term is produced by multiplying the preceding term by

r

. This makes the constant-ratio structure explicit but requires computing all earlier terms to reach a later one.

Converting between forms works the same way as for arithmetic sequences. Applying the recursive rule

n - 1

times gives

a_n = c \cdot r^{n-1}

, recovering the explicit formula. In the other direction, dividing

a_n = a_1 r^{n-1}

a_{n-1} = a_1 r^{n-2}

yields

\frac{a_n}{a_{n-1}} = r

, which gives the recursive rule

a_n = r \cdot a_{n-1}

.

Both representations define the same sequence. The explicit form is better for computing isolated terms; the recursive form emphasizes the multiplicative process and connects to models where each step scales the previous state by a fixed factor.

Finite Geometric Series

The sum of the first

n

terms of a geometric sequence is:

S_n = a_1 + a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1}

The closed form is:

S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1)

The derivation uses a telescoping trick. Multiply

S_n

r

rS_n = a_1 r + a_1 r^2 + \cdots + a_1 r^n

Subtract from the original sum:

S_n - rS_n = a_1 - a_1 r^n

Factor:

S_n(1 - r) = a_1(1 - r^n)

Divide by

1 - r

to isolate

S_n

. When

r = 1

, every term equals

a_1

and

S_n = na_1

.

For example, the sum

1 + 2 + 4 + 8 + \cdots + 2^9

has

a_1 = 1

r = 2

, and

n = 10

S_{10} = 1 \cdot \frac{1 - 2^{10}}{1 - 2} = \frac{1 - 1024}{-1} = 1023

Infinite Geometric Series

When

|r| < 1

, the terms of a geometric sequence shrink toward zero, and the partial sums

S_n

approach a finite limit as

n \to \infty

. Since

r^n \to 0

, the finite sum formula

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}

simplifies to:

S = \frac{a_1}{1 - r} \quad (|r| < 1)

The series

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots

has

a_1 = \frac{1}{2}

and

r = \frac{1}{2}

, so

S = \frac{1/2}{1 - 1/2} = 1

. Adding half of the remaining distance at each step eventually fills the full unit.

Repeating decimals provide another illustration. The decimal

0.333\ldots = \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots

is a geometric series with

a_1 = \frac{3}{10}

and

r = \frac{1}{10}

, summing to

\frac{3/10}{9/10} = \frac{1}{3}

.

When

|r| \geq 1

, the terms do not diminish (or grow in magnitude), and the partial sums grow without bound. The series diverges and no finite sum exists. The boundary case

r = -1

produces partial sums that oscillate between

a_1

and

0

without settling — this too is divergent.

Geometric Mean

The geometric mean of two positive numbers

a

and

b

is:

G = \sqrt{ab}

In a geometric sequence with positive terms, every term is the geometric mean of its two neighbors:

a_n = \sqrt{a_{n-1} \cdot a_{n+1}}

This follows from the constant-ratio property:

a_{n-1} = \frac{a_n}{r}

and

a_{n+1} = a_n r

, so

a_{n-1} \cdot a_{n+1} = a_n^2

.

Inserting geometric means between two positive values

a

and

b

requires placing

k

terms so that the resulting

k + 2

values form a geometric sequence. The common ratio must be

r = \left(\frac{b}{a}\right)^{1/(k+1)}

, and the inserted terms are

ar, ar^2, \ldots, ar^k

.

The AM–GM inequality states that for any set of positive numbers, the geometric mean never exceeds the arithmetic mean:

G \leq A

Equality holds only when all values are identical. This inequality extends to include the harmonic mean:

H \leq G \leq A