Geometric Sequences

Q: Can the common ratio be negative?

Yes. A sequence like $2, -6, 18, -54, \ldots$ has $r = -3$, and the terms alternate sign forever. It's still geometric — dividing any term by the previous one gives $-3$ every time. The common difference of an arithmetic sequence can be negative too; that just means the sequence decreases.

A sequence is just an ordered list of numbers, and the two kinds you'll meet constantly are arithmetic and geometric. An arithmetic sequence adds the same amount each step: $5, 8, 11, 14, \ldots$ adds $3$ every time. A geometric sequence multiplies by the same amount each step: $5, 10, 20, 40, \ldots$ doubles every time. That one distinction — add versus multiply — drives everything else about them.

The payoff is the explicit formula, which jumps straight to any term without listing all the ones before it. For arithmetic sequences it's $a_n = a_1 + (n - 1)d$ ; for geometric it's $a_n = a_1 \cdot r^{\,n-1}$ . Learn what each letter means and you can find the $50$ th term as fast as the $5$ th.

Arithmetic sequences: a common difference

In an arithmetic sequence, each term is the previous term plus a fixed number called the common difference, written $d$ . Find $d$ by subtracting any term from the one after it: in $7, 11, 15, 19, \ldots$ the difference is $11 - 7 = 4$ . If the sequence decreases, $d$ is negative.

The explicit formula is $a_n = a_1 + (n - 1)d$ , where $a_1$ is the first term and $n$ is the term number. The logic: to reach the $n$ th term you start at $a_1$ and take $n - 1$ steps of size $d$ . The exponent-free structure means arithmetic sequences grow linearly — plot the terms of $2, 5, 8, 11, \ldots$ below and they sit right on a straight line.

Geometric sequences: a common ratio

In a geometric sequence, each term is the previous term times a fixed number called the common ratio, written $r$ . Find $r$ by dividing any term by the one before it: in $3, 6, 12, 24, \ldots$ the ratio is $\dfrac{6}{3} = 2$ . A ratio between $0$ and $1$ makes the sequence shrink; a negative ratio makes the signs alternate.

The explicit formula is $a_n = a_1 \cdot r^{\,n-1}$ . Same logic as before: to reach the $n$ th term you start at $a_1$ and multiply by $r$ a total of $n - 1$ times. The exponent is $n - 1$ , not $n$ — that off-by-one is the single most common mistake on these problems. Geometric sequences grow (or decay) exponentially, so the terms curve upward or downward fast. Plotted below, the geometric sequence $2, 4, 8, 16, \ldots$ bends upward instead of following a line.

How to tell which one you have

Run both tests. Subtract consecutive terms — if you get the same number every time, it's arithmetic. Divide consecutive terms — if you get the same number every time, it's geometric. A sequence like $2, 5, 10, 17, \ldots$ fails both tests (the differences are $3, 5, 7$ ), so it's neither.

Context gives it away too. Situations that add a fixed amount per step are arithmetic: a theater where each row has $3$ more seats than the row in front of it. Situations that scale by a fixed percent or factor per step are geometric: a ball that rebounds to $60\%$ of its previous height on every bounce.

Worked examples

Example 1: an arithmetic term far down the list

Find the $20$ th term of the sequence $7, 11, 15, \ldots$

Find the common difference

d = 11 - 7 = 4

Write the explicit formula

a_n = a_1 + (n - 1)d

Substitute

a_1 = 7

d = 4

n = 20

a_{20} = 7 + (19)(4)

Simplify

a_{20} = 7 + 76 = 83

Answer: $a_{20} = 83$

Example 2: a geometric term

Find the $8$ th term of the sequence $3, 6, 12, \ldots$

Find the common ratio

r = \dfrac{6}{3} = 2

Write the explicit formula

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

Substitute

a_1 = 3

r = 2

n = 8

— the exponent is

7

, not

8

a_8 = 3 \cdot 2^7

Simplify

a_8 = 3 \cdot 128 = 384

Answer: $a_8 = 384$

Example 3: an arithmetic sequence in context

A theater has $14$ seats in the first row, and each row behind it has $3$ more seats than the row in front. How many seats are in row $25$ ?

Adding a fixed

3

seats per row makes this arithmetic

a_1 = 14, \quad d = 3

Write the formula for row

n

a_n = 14 + (n - 1)(3)

Substitute

n = 25

a_{25} = 14 + (24)(3)

Simplify

a_{25} = 14 + 72 = 86

Answer: Row $25$ has $86$ seats

Example 4: a shrinking geometric sequence

Find the $6$ th term of the sequence $160, 80, 40, \ldots$

Find the common ratio

r = \dfrac{80}{160} = \dfrac{1}{2}

Write the explicit formula

a_n = 160 \cdot \left(\dfrac{1}{2}\right)^{n-1}

Substitute

n = 6

a_6 = 160 \cdot \left(\dfrac{1}{2}\right)^{5}

Simplify

a_6 = \dfrac{160}{32} = 5

Answer: $a_6 = 5$

Try one yourself

Common questions

Why is the exponent

n - 1

and not

n

Because the first term uses zero multiplications. By the time you reach term $n$ , you've multiplied by $r$ only $n - 1$ times — once for each step after the first term. Same idea in the arithmetic formula: term $n$ is $n - 1$ steps of size $d$ past $a_1$ . Test any formula on $n = 1$ ; it should return $a_1$ exactly.

Can the common ratio be negative?

Yes. A sequence like $2, -6, 18, -54, \ldots$ has $r = -3$ , and the terms alternate sign forever. It's still geometric — dividing any term by the previous one gives $-3$ every time. The common difference of an arithmetic sequence can be negative too; that just means the sequence decreases.

What if a sequence is neither arithmetic nor geometric?

Then neither formula applies, and that's common — the perfect squares $1, 4, 9, 16, \ldots$ are neither, since their differences grow and their ratios shrink. Always verify the pattern with at least two differences or two ratios before committing to a formula.

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Geometric Sequences

Arithmetic sequences: a common difference

Geometric sequences: a common ratio

How to tell which one you have

Worked examples

Example 1: an arithmetic term far down the list

Example 2: a geometric term

Example 3: an arithmetic sequence in context

Example 4: a shrinking geometric sequence

Try one yourself

Common questions

Keep learning

Exponential Growth and Decay

Introduction to Functions

Product & Quotient Rules