Geometric Sequence Calculator — Finite and Infinite Sums

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 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).

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.

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.