What is the formula for a geometric sequence?

The n-th term formula is a sub n equals a sub 1 times r raised to the power n minus 1, where a sub 1 is the first term and r is the common ratio. The sum of the first n terms is given by S sub n equals a sub 1 times the quantity r to the n minus 1, divided by r minus 1, valid for r not equal to 1. When r equals 1, the sum reduces to n times a sub 1.

What is the infinite sum of a geometric series?

If the absolute value of the common ratio r is less than 1, the infinite geometric series converges to S infinity equals a sub 1 divided by the quantity 1 minus r. For absolute value of r greater than or equal to 1, the series diverges. The convergent case is fundamental in calculus, probability, and many applications.

How do you find the common ratio?

Divide any term by the previous one. If a sub k and a sub k plus 1 are consecutive terms with a sub k not zero, then r equals a sub k plus 1 divided by a sub k. If you know two terms a sub m and a sub n with m less than n, then r raised to the power n minus m equals a sub n divided by a sub m, so r equals the quantity a sub n divided by a sub m, raised to the power 1 divided by n minus m.

How is a geometric sequence different from an arithmetic sequence?

An arithmetic sequence adds a constant common difference d at each step; a geometric sequence multiplies by a constant common ratio r at each step. Arithmetic grows linearly. Geometric grows or decays exponentially. The two are the fundamental building blocks of sequence theory in algebra.

Geometric Sequence Calculator — Finite and Infinite Sums

Q: What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term equals the previous term multiplied by a constant called the common ratio. Starting from a first term a sub 1 and multiplying by the common ratio r each step, the n-th term is given by a sub n equals a sub 1 times r raised to the power n minus 1. Examples include 2, 6, 18, 54, 162 (with r equals 3) and 100, 50, 25, 12.5, 6.25 (with r equals one half).

Setting the Parameters

DemoTwo new parameter pairs, then the opening pair again

Step 0 of 3

The card opens with the Sequence drop-down set to Geometric sequence; the other entry switches the card to the Arithmetic sequence, which has its own page, Arithmetic sequence calculator. Under it the Parameters row holds the two numbers that fix the sequence, First term a₁ and Common ratio r, opening with

a_1 = 2

and

r = 3

. Type new values and click Apply or press Enter; every tab recomputes at once with the rule

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

. With

a_1 = 1

and

r = 2

the first eight terms are the powers of two from

1

to

128

with the sum

255

; with

a_1 = 3

and

r = 0.5

they halve from

3

to

0.0234375

and the sum is

5.9765625

.

Both boxes take any decimal, negative values included; a blank or non-numeric box is refused with "Enter a number in every parameter box." and the tabs keep their last values. Which two numbers define the sequence is explained in what is a geometric sequence.

Listing the First N Terms

DemoThree counts in List first N

Step 0 of 3

The explorer opens on the List first N tab, one of five tabs across the top of the card; the small ? beside each tab shows what the mode does. Type how many terms you want in the Show first box and click Update or press Enter. Three cards answer at once: Showing gives the count, Largest the last term with its index, and Sum the total of the listed terms,

S_N = a_1 \frac{r^N - 1}{r - 1}

. Under the cards every term is laid out as a tile with its index above its value.

With the opening parameters the box accepts

1

to

100

:

8

gives

a_8 = 4,374

with the sum

6,560

,

12

gives

a_{12} = 354,294

with the sum

531,440

, and the box is capped at

100

terms; a single term gives

a_1 = 2

. Large values are printed with thousands separators. A count of 0, a decimal or an empty box is refused with "Enter a positive integer." There is no Reset button: type a new count and click Update again. The panel on the right, List first N, states the rule the tiles come from, and its walkthrough repeats the last term and the sum with your own

N

. The rule is explained in the n-th term formula and the sum in the finite sum formula.

Listing a Range of Indexes

DemoThe range tab, a new range, two bad ranges

Step 0 of 4

Click the List a – b tab to list the terms between two positions instead of starting from

1

. Type the From index and the To index and click List range or press Enter. The cards give the Count

b - a + 1

, the Smallest term

a_a

and the Largest term

a_b

, and the tiles below list every term in between with its index. The tab opens with

1

to

6

, 6 terms from

a_{1} = 2

to

a_{6} = 486

; indexes

4

to

9

give 6 terms from

54

to

13,122

.

Both boxes must hold positive integers, the first no larger than the second, and the range may hold at most

100

terms; otherwise a red box explains: "Enter two positive integer indexes", "From-index must be less than or equal to to-index", or "Range too wide — cap is 100 terms". The panel on the right, List a – b, explains the mode, and its walkthrough names the smallest and largest term of your range. The rule each tile uses is explained in the n-th term formula.

Testing a Number

DemoThe test tab, a non-member, a member, a decimal

Step 0 of 4

Click the Test a number tab to ask whether a given number is a term of the sequence. Type it in the Test number box and click Check or press Enter. A green box with a tick answers yes and names the index, as for the opening value

162

, which is

a_{5}

; a red box with a cross answers no, as for

100

, and the walkthrough on the right then shows the two nearest terms,

a_{4} = 54

and

a_{5} = 162

. A large value is handled the same way:

39,366

is

a_{10}

.

The box takes any number, decimals and negatives included, since the parameters may be decimals too; an empty box is refused with "Enter a number". The test is exact: the explorer multiplies

a_1

by

r

step by step and stops as soon as the running term reaches

m

or passes it. The panel on the right, Test a number, states this rule, and its walkthrough carries out the arithmetic for your number. The formula behind it is in the n-th term formula.

Looking Up a Term by Index

DemoThe lookup tab, two indexes, n = 0

Step 0 of 4

Click the Lookup by index tab to compute one term directly. Type the position in the Index n box and click Get term or press Enter. Two cards answer: Index repeats

n

and Value gives

a_n

with its index as a subscript. The tab opens with

n = 10

, which gives

a_{10} = 39,366

;

n = 15

gives

9,565,938

and

n = 20

gives

2,324,522,934

. The box takes positive integers only;

0

is refused with "Enter a positive integer index".

The panel on the right has two parts in every mode. How it works names the rule the mode uses, here

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

, and links to the section of this page that explains it; Walkthrough applies the rule to the numbers you typed, so for the lookup it shows the arithmetic with your

n

and the current parameters filled in. Both change as soon as you switch tabs or click a button. The rule and where it comes from are in the n-th term formula.

Using the Solver Tab

DemoThe solver tab: a term, an unknown moved, a sum

Step 0 of 4

Click the Solver tab, the fifth tab, to work the formulas backwards. Three buttons choose the relation: Find term (aₙ) uses

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

, Sum Sₙ uses

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

and Infinite sum S∞ uses

S_\infty = \frac{a_1}{1 - r}

for

|r| < 1

. The first two have four boxes and the third has three; the one marked solve with a ? is the unknown and the others must be filled. The unknown opens as

a_n

,

S_n

or

S_\infty

; click the solve radio beside another box to make that the unknown. Click Solve or press Enter and the answer appears under the grid, while the walkthrough on the right shows the substitution.

With

a_1 = 2

,

r = 3

and

n = 10

the term is

a_{10} = 39,366

; with

n = 5

the sum is

S_5 = 242

; with

a_1 = 2

and

r = 0.5

the infinite sum is

4

, while

r = 3

is refused with "Series diverges when |r| ≥ 1". Switching the relation clears the boxes, and clicking Solve with a box empty gives "Fill all fields except the one being solved for". Solving

S_n

for

r

is not supported, and solving for

n

needs

r > 0

. The solver boxes are independent of the Parameters row. Every relation is explained in using the solver — three modes.

What is a Geometric Sequence?

A geometric sequence (also called a geometric progression, GP) is a sequence where each term is obtained by multiplying the previous term by a constant. The constant is called the common ratio, denoted

r

.

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

Examples.

•

a_1 = 2

,

r = 3

: gives

2, 6, 18, 54, 162, 486, \ldots

— exponential growth.
•

a_1 = 100

,

r = 1/2

: gives

100, 50, 25, 12.5, 6.25, 3.125, \ldots

— exponential decay.
•

a_1 = 1

,

r = 2

: gives

1, 2, 4, 8, 16, 32, \ldots

— powers of 2.
•

a_1 = 1

,

r = -1

: gives

1, -1, 1, -1, \ldots

— alternating signs.
•

a_1 = 5

,

r = 0

: gives

5, 0, 0, 0, \ldots

— degenerate after the first term.

Three key formulas.

• The

n

-th term:

a_n = a_1 r^{n - 1}

.
• Sum of first

n

terms:

S_n = a_1 \dfrac{r^n - 1}{r - 1}

for

r \neq 1

;

S_n = n a_1

for

r = 1

.
• Infinite sum:

S_\infty = \dfrac{a_1}{1 - r}

when

|r| < 1

; diverges otherwise.

Parametric nature. Like arithmetic sequences, geometric sequences depend on two parameters:

a_1

and

r

. The explorer lets you set both and explore the resulting sequence and sums.

For deeper coverage see geometric sequence, geometric series, and exponential function.

The definition is explained in Geometric sequences: definition, and the overview page compares the main types in Geometric sequences.

Finding terms works the formula through more examples, and Recursive formula gives the step-by-step form

a_n = r \cdot a_{n-1}

.

Finite geometric series proves the formula and applies it.

Infinite geometric series explains convergence and the limit in detail.

Each relation the solver uses is derived in Finding terms, Finite geometric series and Infinite geometric series.

The n-th Term Formula

Derivation. Starting from

a_1

, multiply by

r

once to get

a_1 r

, twice to get

a_1 r^2

. After

n - 1

multiplications, the

n

-th term is

a_n = a_1 r^{n - 1}

Inverting the formula. Given

a_n

,

a_1

, and

r

, solve for

n

using logarithms:

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

This works only when

r > 0

,

r \neq 1

,

a_1 \neq 0

, and

a_n / a_1 > 0

. Negative

r

requires complex logarithms;

r = 1

makes every term equal to

a_1

and

n

is undetermined.

Examples.

• Sequence

3, 6, 12, 24, \ldots

:

a_1 = 3

,

r = 2

. The 10th term:

a_{10} = 3 \cdot 2^9 = 3 \cdot 512 = 1536

.
• Sequence

1000, 500, 250, \ldots

:

a_1 = 1000

,

r = 1/2

. The 10th term:

a_{10} = 1000 \cdot (1/2)^9 = 1000/512 \approx 1.95

.
• Membership test: is 162 in the sequence

2, 6, 18, 54, \ldots

(with

r = 3

)? Solve

162 = 2 \cdot 3^{n-1} \implies 81 = 3^{n-1} \implies n - 1 = 4 \implies n = 5

. Yes —

162 = a_5

.

Two-parameter recovery. Given two terms

a_m

and

a_n

with

m < n

:

r^{n - m} = a_n / a_m

, so

r = (a_n / a_m)^{1/(n - m)}

. Then

a_1 = a_m / r^{m - 1}

.

Sign of $r$ . Negative

r

produces alternating-sign sequences:

a_1 = 1, r = -2

gives

1, -2, 4, -8, 16, -32, \ldots

. The closed form still applies; signs alternate based on parity of

n - 1

.

The Finite Sum Formula

Finite geometric sum. 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} = a_1 \frac{r^n - 1}{r - 1} \quad (r \neq 1)

For

r = 1

, every term equals

a_1

, and

S_n = n a_1

.

Derivation. Write the sum:

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

Multiply both sides by

r

:

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

Subtract:

S_n - r S_n = a_1 - a_1 r^n \implies S_n (1 - r) = a_1 (1 - r^n) \implies S_n = a_1 \frac{1 - r^n}{1 - r} = a_1 \frac{r^n - 1}{r - 1}

Examples.

•

a_1 = 2

,

r = 3

,

n = 5

:

S_5 = 2 \cdot (3^5 - 1)/(3 - 1) = 2 \cdot 242/2 = 242

. Verify:

2 + 6 + 18 + 54 + 162 = 242

. ✓
•

a_1 = 1

,

r = 1/2

,

n = 10

:

S_{10} = (1/2^{10} - 1)/(1/2 - 1) = (1/1024 - 1)/(-1/2) = (1023/1024) \cdot 2 = 2046/1024 \approx 2.0

.

Solving for $n$ . Given

S_n

,

a_1

, and

r

, the formula is invertible:

r^n = \frac{S_n (r - 1)}{a_1} + 1 \implies n = \frac{\ln\left(\dfrac{S_n (r - 1)}{a_1} + 1\right)}{\ln r}

This requires

r > 0, r \neq 1

, and the argument of the logarithm to be positive.

Solving for $r$ . Given

S_n

,

a_1

, and

n

, the equation

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

is transcendental in

r

— no closed-form solution in general. The solver flags this as unsupported.

The Infinite Sum and Convergence

Infinite geometric series. What happens as

n \to \infty

?

S_\infty = a_1 + a_1 r + a_1 r^2 + \cdots = \lim_{n \to \infty} a_1 \frac{r^n - 1}{r - 1}

Convergence condition. The series converges if and only if

|r| < 1

. When it converges:

S_\infty = \frac{a_1}{1 - r}

Why $|r| < 1$ . As

n \to \infty

,

r^n \to 0

when

|r| < 1

, giving the formula. When

|r| \geq 1

,

r^n

does not converge (it grows, oscillates, or stays bounded but doesn't approach zero), and the series diverges.

Classic examples.

• Zeno's paradox.

1 + 1/2 + 1/4 + 1/8 + \cdots = \dfrac{1}{1 - 1/2} = 2

. The pieces sum to a finite total despite there being infinitely many.
• Repeating decimals.

0.\overline{3} = 0.333\ldots = \dfrac{0.3}{1 - 0.1} = \dfrac{1}{3}

. Every repeating decimal is a geometric series.
• Convergent but slowly.

a_1 = 1, r = 0.99

:

S_\infty = 100

. The sequence approaches zero so slowly that it takes thousands of terms to get close.

Diverging cases.

•

r = 1

:

S_n = n a_1 \to \infty

(unless

a_1 = 0

).
•

r = -1

:

S_n

oscillates between

0

and

a_1

, never converging.
•

|r| > 1

:

|S_n|

grows without bound.

The geometric series test. The convergence behavior of

\sum r^n

is used as a benchmark for testing other series: comparison test, ratio test, root test all reference geometric series.

Power series. The geometric series is the simplest power series.

\sum r^n = 1/(1 - r)

for

|r| < 1

extends to functions:

1/(1 - x) = 1 + x + x^2 + x^3 + \cdots

for

|x| < 1

. The cornerstone of formal power series and generating functions.

Using the Solver — Three Modes

The geometric-sequence explorer includes a solver mode with three sub-modes:

• Find term ( $a_n$ ) — uses

a_n = a_1 r^{n - 1}

. Fill any three of

\{a_1, r, n, a_n\}

and the solver computes the fourth.
• Sum $S_n$ — uses

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

for

r \neq 1

. Fill any three of

\{a_1, r, n, S_n\}

and the solver computes the fourth (except solving for

r

, which is transcendental and not supported).
• Infinite sum $S_\infty$ — uses

S_\infty = a_1/(1 - r)

for

|r| < 1

. Fill any two of

\{a_1, r, S_\infty\}

and the solver computes the third, flagging cases where

|r| \geq 1

as divergent.

Solving for $a_n$ . Direct:

a_n = a_1 r^{n - 1}

.

Solving for $a_1$ . Rearrange:

a_1 = a_n/r^{n - 1}

(term sub-mode),

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

(sum sub-mode), or

a_1 = S_\infty(1 - r)

(infinite sum).

Solving for $r$ .

(n - 1)

-th root:

r = (a_n/a_1)^{1/(n - 1)}

(term sub-mode). For infinite sum:

r = 1 - a_1/S_\infty

(with a convergence check). For finite sum: transcendental — not supported.

Solving for $n$ . Logarithm:

n = \ln(a_n/a_1)/\ln r + 1

(term sub-mode);

n = \ln(S_n(r - 1)/a_1 + 1)/\ln r

(sum sub-mode). Both require

r > 0

,

r \neq 1

.

Edge cases the solver handles.

•

r = 1

: term-sub-mode has every term equal; sum sub-mode reduces to

S_n = n a_1

.
•

r \leq 0

in solve-for-

n

: real logarithm doesn't exist, the solver flags this.
•

|r| \geq 1

in infinite sum: divergence flagged.
•

a_1 = 0

: most equations become degenerate; flagged.

Practical use. Solve word problems: "An investment of \

1000 grows by 5% per year. After how many years does it reach \

2000?" → solve for

n

given

a_1 = 1000

,

r = 1.05

,

a_n = 2000

.

Common Applications

Geometric sequences model exponential growth and decay across science, finance, and computing.

• Compound interest. \

1000 at 5% annual interest, compounded yearly: balance follows the geometric sequence

1000, 1050, 1102.50, \ldots

with

r = 1.05$.
• Population growth. Bacterial population doubles every hour:

r = 2

. After

n

hours, population is

a_1 \cdot 2^{n - 1}

relative to the starting count.
• Radioactive decay. Mass remaining after each half-life follows

r = 1/2

. After

n

half-lives,

a_1 / 2^n

mass remains.
• Mortgage payments. Outstanding balance after each payment follows a geometric-related sequence (with adjustments for the principal payment). The total payment formula is the geometric sum.
• Algorithm analysis. Binary search splits the search space by half each step — a geometric sequence with

r = 1/2

. Total steps:

\log_2 n

.
• Fractals and self-similarity. The Koch snowflake's perimeter at each iteration multiplies by

4/3

: a geometric sequence with

r = 4/3

that grows without bound.
• Repeating decimals as fractions.

0.\overline{abc} = abc/999

, derived from the geometric-series formula.
• Pyramid schemes. Each level multiplies participants by some ratio

r

. Total participants grow exponentially until the scheme exceeds the population.
• Music tuning. Equal-temperament tuning divides the octave into 12 equal ratios, each

\sqrt[12]{2} \approx 1.0595

. Frequencies follow a geometric sequence.
• Light absorption. Light passing through a medium of uniform absorption loses a fixed fraction per unit length — Beer's law, an exponential decay (geometric in the discrete approximation).

Identifying geometric sequences shows how to spot a constant ratio in data; Geometric mean is the average that fits such data.

Common Mistakes

• Off-by-one in the $n$ -th term formula.

a_n = a_1 r^{n - 1}

, not

a_1 r^n

. The 1st term is

a_1

(zero multiplications), the 2nd is

a_1 r

(one multiplication).

• Wrong sum formula.

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

or equivalently

a_1 (1 - r^n)/(1 - r)

. Both are correct; common errors include using

n

as an exponent on

r - 1

or forgetting to divide by

r - 1

.

• Treating $r = 1$ with the general formula. When

r = 1

, the formula has

0/0

— undefined. The correct value is

S_n = n a_1

. Always check

r = 1

as a special case.

• Forgetting the convergence condition for $S_\infty$ . The infinite sum exists only when

|r| < 1

. Applying

a_1/(1 - r)

when

|r| \geq 1

gives a meaningless number. The classic error: writing

1 + 2 + 4 + 8 + \cdots = 1/(1 - 2) = -1

. The series diverges; the formula doesn't apply.

• Confusing arithmetic and geometric. Arithmetic adds; geometric multiplies. "Increases by 10% per year" is geometric (

r = 1.1

); "increases by \

10 per year" is arithmetic (

d = 10$).

• Negative $r$ and parity. With

r < 0

, terms alternate in sign.

a_1 = 1, r = -2

gives

1, -2, 4, -8, 16

. The sum formula still works but the result oscillates as

n

grows.

• Logarithm restrictions in solve-for- $n$ .

n = \ln(a_n/a_1)/\ln r + 1

requires positive arguments. Negative or zero

r

, or

a_n

and

a_1

with opposite signs, makes the formula fail. The solver flags these.

• Floating-point errors with non-integer ratios.

r = 1/3

is exact in mathematics but approximate in floating-point. Computing

a_{100}

with

r = 1/3

accumulates rounding error. Use symbolic computation or rational arithmetic for exact results.

• Conflating geometric series with arithmetic series for ratio. "Common ratio" is for geometric; "common difference" is for arithmetic. The terms are not interchangeable.

The one-page summary Geometric sequences at a glance is a quick check against these slips.

Related Sequences and Concepts

Arithmetic Sequence — adds a constant:

a_n = a_1 + (n - 1) d

. Geometric's linear counterpart.

Geometric Series — the sum of a geometric sequence. Both finite and infinite versions are central.

Exponential Function — the continuous analog.

f(x) = a_1 r^x

. Geometric sequences are exponentials sampled at integer points.

Power Series — the generalization to varying coefficients:

\sum c_n x^n

. The geometric series

\sum r^n = 1/(1 - r)

is the simplest non-trivial power series.

Convergence — the criterion for infinite-series sums.

|r| < 1

for geometric is the prototype.

Ratio Test — convergence test for series, comparing consecutive terms' ratio to a geometric series benchmark.

Compound Interest — the canonical finance application. Future value follows a geometric sequence with

r = 1 + i

where

i

is the per-period rate.

Exponential Growth and Decay — the natural-science applications. Bacterial growth, radioactive decay, drug elimination — all modeled as geometric sequences (discrete) or exponentials (continuous).

Logarithm — the inverse of exponential. Solving for

n

in a geometric sequence uses logarithms.

Common Ratio — the parameter

r

. Determines whether the sequence grows, decays, oscillates, or stays constant.

Geometric Mean — the average. For a geometric sequence, the geometric mean of any two terms equals the term between them:

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

for any

n \geq 2

.

Repeating Decimal — every repeating decimal is a geometric series, converting to a rational number via the infinite-sum formula.

Geometric Progression — alternate name for geometric sequence. Used interchangeably.

Linear Recurrence (homogeneous) — the general class. Geometric sequences are first-order linear recurrences

a_n = r \cdot a_{n-1}

.

Half-Life — the time for a quantity to halve. A geometric sequence with

r = 1/2

measured at half-life intervals.

Doubling Time — the time for a quantity to double. A geometric sequence with

r = 2

at doubling-time intervals.

Beer's Law — light absorption following exponential (geometric in discrete steps) decay through a medium.

Self-Similarity and Fractals — geometric scaling underlies fractal structures; iteration counts follow geometric sequences.

Arithmetic sequences and Harmonic sequences are introduced on the sequences overview page; the arithmetic case has its own lesson in Arithmetic sequences: definition.