Geometric Sequences & Series Formula

Definition

Two patterns start at 10: one adds 10 at every step, the other doubles at every step.

The first gives

10, 20, 30, 40, \ldots

, the second

10, 20, 40, 80, \ldots

. After two steps they agree; after that they part company fast.

By step 10 the adding pattern has reached 100 and the doubling pattern 5120.

step n	adding 10	its differences	doubling	its ratios
1	10		10
2	20	10	20	2
3	30	10	40	2
4	40	10	80	2
5	50	10	160	2
10	100	10	5120	2

The adding pattern keeps a constant difference; the doubling pattern keeps a constant ratio.

Dividing each doubling term by the one before always gives

2

, even though the differences

10, 20, 40, 80

keep growing. A constant quotient, rather than a constant gap, is what holds the second pattern together.

The definition

A geometric sequence is a sequence in which each term divided by the term before it is the same nonzero constant

r

, the common ratio.

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

r

:

Common Ratio

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

Learn more about this formula: Common Ratio →

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:

General Term (Geometric Sequence)

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

Learn more about this formula: General Term (Geometric Sequence) →

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So

10, 20, 40, 80, \ldots

is geometric with

r = 2

, while

10, 20, 30, 40, \ldots

is arithmetic, not geometric.

The ratio decides the shape: growth, decay or alternation — always exponential, never a straight line.

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

.

Three ratios, three shapes

The section’s three sequences plotted against n. With r = 3 the dots 2, 6, 18, 54 climb ever faster. With r = 1/3 the dots 81, 27, 9, 3, 1 fall toward the axis without reaching it. With r = −2 the dots 4, −8, 16, −32 jump from one side of the axis to the other, each farther out than the last. The sign of r controls the side, its size controls growth or decay.

With −1 < r < 0 the jumps shrink instead: the terms alternate and decay toward 0.

Range of r	Term behavior	Series behavior	Example
r > 1	same-sign terms grow without bound	diverges	2, 6, 18, 54, … (r = 3)
0 < r < 1	same-sign terms decay toward 0	converges to a₁ ⁄ (1 − r)	81, 27, 9, 3, 1, … (r = 1⁄3)
−1 < r < 0	signs alternate, magnitudes decay toward 0	converges to a₁ ⁄ (1 − r)	8, −4, 2, −1, 0.5, … (r = −1⁄2)
r < −1	signs alternate, magnitudes grow unbounded	diverges	4, −8, 16, −32, … (r = −2)
r = 1	constant: every term equals a₁ (also arithmetic with d = 0)	diverges: S_n = n · a₁	5, 5, 5, 5, …
r = −1	oscillates between ±a₁ at constant magnitude	diverges: partial sums oscillate, never settle	7, −7, 7, −7, …

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

.

Zero cannot appear: a ratio needs a nonzero term to divide by.

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.

How to test a list of numbers for a constant ratio is covered in identifying geometric sequences.

The rest of the page covers identifying geometric sequences, finding terms, the recursive formula, finite and infinite geometric series, and the geometric mean.

Geometric Sequence Notation

Notation

Geometric Sequence Notation

A letter for the ratio, a subscript that counts the terms summed, a bare letter once the sum runs forever, and a letter for the middle value.

The subscript as position, the brace form and sigma are set out at sequence notation; the root in the geometric mean follows radical notation.

r = \dfrac{a_{n+1}}{a_n}

r, the common ratio

The constant factor from each term to the next — the multiplying twin of the common difference

d

. It enters the general term as a power:

a_n = a_1 r^{n-1}

, one less than the position.

Cases

r < 0

alternates the signs,

0 < |r| < 1

shrinks the terms.

r = 0

and

a_1 = 0

are excluded, so the ratio is always defined.

Also written

q

— the usual letter in many European textbooks.

Do not confuseThe exponent count.

a_{10} = a_1 r^{9}

, not

a_1 r^{10}

— the first term already holds place

1

.

S_n

S sub n — the sum of the first n terms

A partial sum: the subscript counts the terms added, not the exponent of the last one.

1 + 2 + \cdots + 2^9

is

S_{10}

— exponents

0

to

9

, ten terms.

CasesThe closed form

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

needs

r \neq 1

; at

r = 1

every term is

a_1

and

S_n = n a_1

.

Also written

\sum_{k=1}^{n} a_1 r^{k-1}

, or

\sum_{k=0}^{n-1} a_1 r^{k}

— the same sum with the index starting at

0

.

Do not confuseThe last exponent. Reading

n = 9

off

2^9

drops a term and gives

511

instead of

1023

.

Same glyph elsewhereOn the square numbers page

S_n

is the n-th square

n^2

— one term, not a sum.

S = \sum_{k=1}^{\infty} a_1 r^{k-1}

S, the sum of the infinite series

The limit of the partial sums:

S = \lim_{n \to \infty} S_n = \frac{a_1}{1 - r}

. No subscript — there is no term count left.

CasesExists only for

|r| < 1

. For

|r| \geq 1

the symbol names nothing: forcing

\frac{a_1}{1 - r}

on

1 + 2 + 4 + \cdots

gives

-1

.

Also written

S_\infty

— common in school texts, the subscript standing for the endless count.

Do not confuseA large

S_n

. Every finite partial sum of

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

falls short of

S = 1

;

S

is the value they approach.

G = \sqrt{ab}

G, the geometric mean of a and b

The middle value that makes

a, G, b

geometric. In a geometric sequence with positive terms, each term is the geometric mean of its two neighbours (section 7).

Cases

G \leq A

, with equality only for equal values; with the harmonic mean the chain is

H \leq G \leq A

.

Also written

\text{GM}

; for

n

numbers,

\sqrt[n]{a_1 a_2 \cdots a_n}

.

Do not confuseThe arithmetic mean

A = \frac{a + b}{2}

, the value the AM–GM inequality compares it with.

All algebra symbols →↑ Sequences

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Notation · Cheat sheet

Sequence Notation at a Glance

The marks from the notation cards across the sequence pages: how each is read and what it means. Every group links to the full card on its page.

Sequences

a_n

a sub n

The term in place

n

; the subscript is a position.

a_1, a_2, a_3, \ldots

a one, a two, a three, and so on

The terms listed in order; the dots need a stated rule.

\{a_n\}_{n=1}^{\infty}

the sequence a sub n

The whole sequence as one object.

Arithmetic sequences

a_{n+1} = a_n + d

next term is this term plus d

The recurrence;

d

is the common difference,

a_1

the start.

\textstyle\sum_{k=1}^{n} a_k

sum of a sub k, k from 1 to n

Sigma: the sum

S_n

of the first

n

terms.

Geometric sequencesThis page

r

r, the common ratio

The factor from each term to the next:

a_n = a_1 r^{n-1}

.

S_n

S sub n

Sum of the first

n

terms;

n

counts terms.

S

S, the infinite sum

Equals

a_1 / (1 - r)

, only when

|r| < 1

.

G

G, the geometric mean

G = \sqrt{ab}

, the middle value of

a, G, b

.

Fibonacci sequence

F_n

F sub n

The

n

-th Fibonacci number, counted from

F_1 = F_2 = 1

.

\phi, \; \psi

phi and psi

The roots of

x^2 = x + 1

, about

1.618

and

-0.618

.

L_n

L sub n

Lucas numbers: the same rule, started at

1, 3

.

Harmonic sequences

a_n = 1/b_n

a sub n is one over b sub n

b_n

is the arithmetic sequence underneath.

H_n

H sub n

The harmonic number

1 + \frac{1}{2} + \cdots + \frac{1}{n}

.

H

H, the harmonic mean

For two numbers,

H = \frac{2ab}{a + b}

.

Triangular numbers

T_n

T sub n

The

n

-th triangular number,

\frac{n(n+1)}{2}

.

T_n = T_{n-1} + n

the one before plus a row of n

The recurrence, from

T_1 = 1

.

\binom{n+1}{2}

n plus one choose two

The same number, counted as pairs.

Square numbers

S_n = n^2

S sub n on this page

The

n

-th square: one term, not a sum.

S_n = S_{n-1} + 2n - 1

the one before plus the next odd number

The recurrence, from

S_1 = 1

.

\textstyle\sum_{k=1}^{n} k^2

sum of k squared

Square each term first, then add.

Prime numbers

p

p, a prime

The letter promises a prime.

p_1, \ldots, p_k

p sub one to p sub k

A list of primes, or the primes of one number.

\pi(n)

pi of n

How many primes are at most

n

.

Same letter, different job

S_n

A sum of

n

terms on the arithmetic and geometric pages; a single square on the square numbers page.

H_n

vs

H

With a subscript, a sum of reciprocals; without one, a mean.

\pi(n)

vs

\pi

A count of primes, not the circle constant.

\{a_n\}

An ordered sequence, not a set: terms keep their order and may repeat.

Sequence symbols table Sequences

Sequence Notation at a Glance·/algebra/sequencesLearn Math Class

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.

THEOREM

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.

Pitfall: Dividing in the wrong order

Dividing each term by the next: for

54, 18, 6, 2

that gives

\frac{54}{18} = 3

, so

r = 3

, as if the sequence grew. The ratio is later over earlier,

\frac{18}{54} = \frac{1}{3}

, so

r = \frac{1}{3}

.

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

.

Pitfall: Using the power n instead of n − 1

Writing

a_{10} = 3 \cdot 2^{10} = 3072

multiplies by

r

once too often, since

a_1

already stands in place

1

. Only

9

steps lead to place

10

:

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}

.

Example

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

.

To find any term of your own geometric sequence, use the geometric sequence calculator.

Given	Find	Equation / approach	Example
a₁, r, n	a_n	a_n = a₁ · rⁿ⁻¹ — substitute directly	a₁ = 3, r = 2 → a₁₀ = 3 · 2⁹ = 1536
a₁, a_n, n	r	r = (a_n ⁄ a₁)^{1 ⁄ (n−1)}	a₁ = 5, a₄ = 40 → r³ = 8 → r = 2
a_n, r, n	a₁	a₁ = a_n ⁄ rⁿ⁻¹	a₆ = 96, r = 2 → a₁ = 96 ⁄ 32 = 3
a₁, r, a value v	n (position of v)	n = 1 + log_r(v ⁄ a₁); use logarithms if powers aren't clean	a₁ = 4, r = 3, v = 2916 → 3ⁿ⁻¹ = 729 → n = 7
Two non-consecutive terms	a₁ and r	divide the two equations to isolate a power of r	a₂ = 6, a₅ = 162 → r³ = 27 → r = 3, a₁ = 2

Recursive Formula

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

Recursive Form (Geometric Sequence)

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

Learn more about this formula: Recursive Form (Geometric Sequence) →

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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}

by

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:

Finite Geometric Series

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

The closed form is:

Finite Geometric Series Sum

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

Learn more about this formula: Finite Geometric Series Sum →

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The derivation uses a telescoping trick. Multiply

S_n

by

r

:

Finite Geometric Series

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

Subtract from the original sum:

Finite Geometric Series

S_n - rS_n = a_1 - a_1 r^n

Factor:

Finite Geometric Series

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

:

Finite Geometric Series

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

Pitfall: Taking n from the last exponent

Reading

n = 9

off the last term

2^9

gives

\frac{1 - 2^9}{1 - 2} = 511

, which leaves out one term. The exponents run from

0

to

9

, so there are

10

terms and

S_{10} = 1023

.

The closed form needs three inputs: the first term, the ratio and the number of terms being added. The first two can be read straight off the sum, while the third has to be counted from the exponents of the terms. In this example the result is one less than

2^{10}

, the power of

2

that would come next, and the same pattern holds for every sum of this kind:

1 + 2 + \cdots + 2^{m} = 2^{m+1} - 1

.

Multiplying a geometric sum by

r

shifts it one place along, so subtracting the two cancels all but two terms.

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:

Infinite Geometric Series Sum

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

Learn more about this formula: Infinite Geometric Series Sum →

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Fact: Achilles catches the tortoise after all

In Zeno's paradox, Achilles runs ten times as fast as a tortoise that starts

100

m ahead, and must first reach where it was, by which time it has moved on, without end. His stages

100 + 10 + 1 + \cdots

form a series with

r = \frac{1}{10}

, so he draws level after exactly

\frac{1000}{9} = 111.\overline{1}

m.

Example

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 comes as close to the full unit as desired, and the infinite sum is exactly

1

.

Repeating decimals provide another illustration.

Example

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}

.

RULE

When

|r| \geq 1

, the terms do not diminish (or grow in magnitude), and the partial sums do not settle to a limit. The series diverges and no finite sum exists.

Pitfall: Summing a series that diverges

Applying

\frac{a_1}{1 - r}

to

1 + 2 + 4 + 8 + \cdots

gives

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

, a negative total for positive terms. With

r = 2

the partial sums

1, 3, 7, 15, \ldots

grow without bound, so there is no sum.

The boundary case

r = -1

produces partial sums that oscillate between

a_1

and

0

without settling — this too is divergent.

Both series summed earlier in this section pass the test:

|r| = \frac{1}{2}

for the halves and

|r| = \frac{1}{10}

for the repeating decimal. For them the partial sums

S_n

settle on a single value, and

\frac{a_1}{1 - r}

is exactly that value. The formula is the limit of the finite sum, so it describes a series only when that limit exists.

Filling the unit by halves

Column n shows the partial sum Sₙ of 1/2 + 1/4 + 1/8 + … in blue and the gap still left below 1 in amber. The gaps are 1/2, 1/4, 1/8, …: each new term covers exactly half of what remains. The blue part never passes the dashed line at 1, but by n = 6 the gap is 1/64 and too thin to see. The partial sums close in on S = 1.

With r = 2 the gap would never close: each term would be larger than everything before it.

Infinitely many terms can have a finite sum, and for a geometric series that happens exactly when

|r| < 1

.

Geometric Mean

The geometric mean of two positive numbers

a

and

b

is:

Geometric Mean

G = \sqrt{ab}

Learn more about this formula: Geometric Mean →

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

Geometric Mean Property

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

Learn more about this formula: Geometric Mean Property →

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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

.

Pitfall: Taking the root by the number of means

For

3

means between

2

and

162

, the cube root

r = \sqrt[3]{81} \approx 4.33

reaches

162

in three steps and leaves only two means. Three means make four steps:

r = \sqrt[4]{81} = 3

, giving

6, 18, 54

.

In this kind of problem the two endpoints are fixed and only the ratio between neighbouring terms is unknown. Once

r

is found, the inserted terms follow by repeated multiplication, and the last step from

ar^k

must land exactly on

b

, which checks the whole list at once.

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

Geometric 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

.

Geometric Sequences at a Glance

The page defined the common ratio, derived the explicit and recursive formulas, the finite-series telescoping sum, the infinite-series convergence rule, and the geometric-mean property. The table below collects each formula with a representative example for quick reference.

Concept	Formula / property	Example
Common ratio	r = a_n+1 ⁄ a_n (constant; r ≠ 0)	5, 15, 45, 135 → r = 3
Explicit (n-th term)	a_n = a₁ · rⁿ⁻¹	a₁ = 3, r = 2 → a_n = 3 · 2ⁿ⁻¹
Recursive	a₁ = c; a_n = r · a_n−1	multiply each term by r to step forward
Finite series sum	S_n = a₁ · (1 − rⁿ) ⁄ (1 − r) for r ≠ 1; if r = 1 then S_n = n · a₁	1 + 2 + 4 + … + 2⁹ = 1023
Infinite series sum	S = a₁ ⁄ (1 − r), valid only when \|r\| < 1	1⁄2 + 1⁄4 + 1⁄8 + … = 1
Geometric-mean property	for positive terms, a_n = √(a_n−1 · a_n+1)	interior term = √(neighbor product)
Inserting k means between a, b	r = (b ⁄ a)^{1 ⁄ (k+1)}; inserts a·r, a·r², …, a·r^k	3 means between 2 and 162 → r = 3 gives 6, 18, 54
AM–GM inequality	G ≤ A (equality iff all values are equal); extends to H ≤ G ≤ A	√(ab) ≤ (a + b) ⁄ 2
Compared with arithmetic	arithmetic = linear (a_n ∝ n); geometric = exponential (a_n ∝ rⁿ)	3, 6, 9, 12 (arithmetic) vs 3, 6, 12, 24 (geometric)

Common Pitfalls

Five mistakes come up again and again with geometric sequences. Each one is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

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1Dividing in the wrong order

When the ratios of consecutive terms are computed, each term is divided by the one after it instead of the one before it. The ratios still come out equal, but each is the reciprocal of the true ratio.

Start

For

54, 18, 6, 2

,

Wrong turn

Error here

\frac{54}{18} = 3, \quad \frac{18}{6} = 3, \quad \frac{6}{2} = 3 \implies r = 3

Right path

The common ratio is

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

, later term over earlier term.

\frac{18}{54} = \frac{1}{3}, \quad \frac{6}{18} = \frac{1}{3}, \quad \frac{2}{6} = \frac{1}{3} \implies r = \frac{1}{3}

Why it happensThe larger number is written on top out of habit, and because all the ratios agree, the constant-ratio test passes and hides the error.

How to checkMultiply the first term by

r

; it must give the second.

54 \cdot 3 = 162

, not

18

, while

54 \cdot \frac{1}{3} = 18

. A sequence of positive terms that shrinks always has

0 < r < 1

.

2Using the power n instead of n − 1

In the general term,

r

is raised to the position

n

instead of to

n - 1

, the number of steps from the first term.

Start

With

a_1 = 3

and

r = 2

,

Wrong turn

Error here

a_{10} = 3 \cdot 2^{10} = 3072

Right path

From place

1

to place

10

there are

9

steps.

a_{10} = 3 \cdot 2^{10 - 1} = 3 \cdot 512 = 1536

The same slip appears when solving for

r

: from

a_1 = 5

and

a_4 = 40

, writing

40 = 5r^4

gives

r^4 = 8

and

r \approx 1.68

, instead of

r^3 = 8

and

r = 2

.

Why it happensThe formula is remembered as "first term times

r

to the

n

", and the fact that

a_1

already occupies the first place, before any multiplication by

r

, gets lost.

How to checkPut

n = 1

into the formula; it must return

a_1

. The wrong version gives

3 \cdot 2^1 = 6

, which is the second term, while

3 \cdot 2^0 = 3

.

3Taking n from the last exponent

In the finite sum formula,

n

is the number of terms added. It is read off the exponent of the last term instead of being counted.

Wrong turn

Error here

For

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

, taking

n = 9

,

S = \frac{1 - 2^9}{1 - 2} = \frac{1 - 512}{-1} = 511

Right path

The exponents run

0, 1, 2, \ldots, 9

, which is

10

terms.

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

Why it happensThe terms are

a_1 r^0, a_1 r^1, \ldots, a_1 r^{n-1}

, so the exponent of the last term is always one less than the number of terms.

How to checkA sum of positive terms is larger than its last term. The wrong answer

511

is smaller than the last term

2^9 = 512

alone, so it cannot be right;

1023 = 511 + 512

is.

4Summing a series that diverges

The infinite-sum formula

S = \frac{a_1}{1 - r}

is applied to a series whose ratio has size

1

or more.

Start

For

1 + 2 + 4 + 8 + \cdots

with

a_1 = 1

and

r = 2

,

Wrong turn

Error here

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

Right path

|r| = 2 \geq 1

, so the terms do not shrink. The partial sums are

1, 3, 7, 15, \ldots

, that is

S_n = 2^n - 1

, and they grow without bound. The series diverges and has no sum.

The same happens at

r = -1

: for

1 - 1 + 1 - 1 + \cdots

the formula gives

\frac{1}{1 - (-1)} = \frac{1}{2}

, but the partial sums alternate

1, 0, 1, 0, \ldots

and never settle.

Why it happensThe formula needs only

a_1

and

r

and returns a number for every ratio except

r = 1

. The condition

|r| < 1

is written beside it, not inside it, and is easy to leave behind.

How to checkCompare the answer with the first few partial sums. A series of positive terms cannot add up to a negative number, and partial sums that keep increasing cannot be approaching

-1

. Checking

|r| < 1

first avoids the problem.

5Taking the root by the number of means

When

k

geometric means are inserted between

a

and

b

, the ratio

\frac{b}{a}

is split into

k

equal factors instead of

k + 1

.

Start

3

means between

2

and

162

,

Wrong turn

Error here

r^3 = \frac{162}{2} = 81

\implies r = \sqrt[3]{81} \approx 4.33

Which gives the list

2, 8.65, 37.44, 162

(values rounded), with only two values between the endpoints.

Right path

3

means and

2

endpoints make

5

terms, and

5

terms have

4

steps between them.

r^4 = \frac{162}{2} = 81

\implies r = \sqrt[4]{81} = 3

The sequence is

2, 6, 18, 54, 162

.

Why it happensThe number of values to insert and the number of steps between neighbouring values are easy to confuse; there is always one more step than there are inserted values.

How to checkCount the values strictly between the endpoints in the finished list; there must be exactly

k

. Equivalently,

a \cdot r^{k + 1}

must equal

b

:

2 \cdot 3^4 = 162

.

Common pitfalls with geometric sequences·/algebra/sequences/geometricLearn Math Class

Four of the five slips miscount or misread the steps between terms: the ratio is a later term over an earlier one, place

n

is reached after

n - 1

multiplications, a sum running from

2^0

to

2^9

has

10

terms, and

k

means make

k + 1

steps. The fifth applies the sum formula

\frac{a_1}{1 - r}

where it does not hold; it needs

|r| < 1

. Putting

n = 1

into a formula, or comparing a sum with its first few partial sums, catches each of them.

Geometric Sequences FAQ

Why does $0.333\ldots$ equal exactly $\frac{1}{3}$ ?

+

Because the decimal is an infinite geometric series in disguise. Writing it out gives

\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots

, with first term

\frac{3}{10}

and ratio

\frac{1}{10}

. Since the ratio is under

1

, the series converges to

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

exactly. It is not an approximation that falls slightly short.Read more →

Why is the condition $|r| < 1$ rather than $r < 1$ ?

+

Because what matters is whether the terms shrink in size, not whether the ratio is small. A ratio of

-\frac{1}{2}

converges perfectly well, since the magnitudes halve each step even as the signs alternate. A ratio of

-2

satisfies

r < 1

yet the terms grow without bound. Absolute value is what captures shrinking.Read more →

What happens to a geometric series when $r = -1$ ?

+

It diverges, though not by growing. The terms are

a_1, -a_1, a_1, -a_1, \ldots

, so the partial sums oscillate between

a_1

and

0

forever without approaching any single value. Divergence means the partial sums fail to settle, which covers oscillation as well as unbounded growth. No finite sum exists.Read more →

Can a sequence be both arithmetic and geometric?

+

Yes, but only a constant one such as

5, 5, 5, 5

, which has

d = 0

and

r = 1

. Any other choice fails: if

a, a + d, a + 2d

were also geometric, the middle term would satisfy

(a + d)^2 = a(a + 2d)

, which simplifies to

d^2 = 0

. So a nonzero common difference always breaks the constant ratio.Read more →

Can a geometric sequence contain zero?

+

No. The first term must be nonzero, and so must the ratio: with

r = 0

every term after the first would be

0

, and the ratio

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

would have nothing to divide by. With

a_1 \neq 0

and

r \neq 0

, every term

a_1 r^{n-1}

is a product of nonzero numbers, so zero never appears.Read more →

Does Achilles ever catch the tortoise?

+

Yes, after a finite distance. In Zeno's paradox Achilles runs ten times as fast as a tortoise that starts

100

m ahead, so each stage of the chase is a tenth of the one before:

100 + 10 + 1 + \cdots

. That is a geometric series with

r = \frac{1}{10}

, and its sum is

\frac{100}{1 - 1/10} = \frac{1000}{9} = 111.\overline{1}

m. Infinitely many stages, but a finite total.Read more →

Geometric Sequences

Constant Ratio

Definition

Geometric Sequence Notation

Geometric Sequence Notation

Identifying Geometric Sequences

Finding Terms

Recursive Formula

Finite Geometric Series

Infinite Geometric Series

Geometric Mean

Geometric Sequences at a Glance

Common Pitfalls

Geometric Sequences FAQ

Why does 0.333…0.333\ldots0.333… equal exactly 13\frac{1}{3}31​?

Why is the condition ∣r∣<1|r| < 1∣r∣<1 rather than r<1r < 1r<1?

What happens to a geometric series when r=−1r = -1r=−1?

Can a sequence be both arithmetic and geometric?

Can a geometric sequence contain zero?

Does Achilles ever catch the tortoise?

Why does $0.333\ldots$ equal exactly $\frac{1}{3}$ ?

Why is the condition $|r| < 1$ rather than $r < 1$ ?

What happens to a geometric series when $r = -1$ ?