What is an arithmetic sequence?

It is a sequence in which the difference between two neighbouring terms is constant; that constant is called the common difference and written r. The terms come from adding the same number: 3, 7, 11, 15 is arithmetic with r = 4. Its nth term is aₙ = a₁ + (n − 1)r.

What is a geometric sequence?

It is a sequence in which the ratio of two neighbouring terms is constant; that constant is called the common ratio and written q. The terms come from multiplying by the same number: 3, 6, 12, 24 is geometric with q = 2. Its nth term is aₙ = a₁ times q to the power of n − 1.

Why does the nth-term formula say n − 1 rather than n?

Because getting from the first term to the nth takes one step fewer than the index says. Reaching a₄ takes three steps, not four. The same correction sits in the exponent of the geometric formula: a₄ = a₁ · q³.

How do I sum the first n terms of an arithmetic sequence?

With Sₙ = (a₁ + aₙ) / 2 · n — the mean of the first and last term multiplied by how many terms there are. If the last term is not given, substitute it from the general formula, which turns the sum into Sₙ = (2a₁ + (n − 1)r) / 2 · n.

How do I sum the first n terms of a geometric sequence?

For q other than 1, with Sₙ = a₁ · (qⁿ − 1) / (q − 1). When q = 1 the sequence is constant and the sum is simply n · a₁ — dividing by q − 1 would then be dividing by zero, which is why that case is always stated separately.

What does it mean that a term is the mean of its neighbours?

In an arithmetic sequence every term but the first is the arithmetic mean of its neighbours: aₙ = (aₙ₋₁ + aₙ₊₁) / 2. A geometric sequence has the multiplicative counterpart: aₙ² = aₙ₋₁ · aₙ₊₁. Both properties are used to insert a missing term between two given ones.

Can the common ratio be negative?

Yes — the terms then alternate in sign, as in 2, −4, 8, −16 with q = −2. It cannot be zero, though, because every later term would be zero and the ratio would stop existing; for the same reason no term of a geometric sequence is ever zero.

Why is compound interest a geometric sequence?

Because after each period the capital is multiplied by the same factor 1 + p, where p is the rate written as a fraction. The balance after successive years is therefore a₁, a₁(1 + p), a₁(1 + p)² — a geometric sequence with ratio 1 + p. Simple interest would give an arithmetic sequence instead, and noticeably less money after a few years.

Intermediate level

Arithmetic and geometric sequences

Two sequences describe almost everything that grows regularly: one keeps adding the same amount, the other keeps multiplying by the same amount. Meet the formulas for the nth term and for the sum of the first n terms of both, the mean property, and the reason a savings account is a geometric sequence.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Where this is used

Real situations where you count exactly the way this lesson teaches:

A savings account
You deposit 10,000 into an account paying 4% a year, compounded annually. The balance after each year is a geometric sequence with ratio q = 1.04: after five years you hold 10,000 · 1.04⁵ = 12,166.53, so 2,166.53 in interest — 166.53 more than plain addition of 400 a year would give. That surplus is exactly the interest earned on interest.
See the lesson: Compound interest
Amphitheatre seating
An architect lays out an auditorium: 14 seats in the first row and 3 more in every row after it. That is an arithmetic sequence with r = 3, so row twelve holds 14 + 11 · 3 = 47 seats and the whole auditorium seats S₁₂ = (14 + 47) / 2 · 12 = 366 people. Adding a thirteenth row buys 50 seats at once — which is why capacity grows faster than the row count.
A loan with falling instalments
A loan of 240,000 repaid in 240 capital instalments of 1,000, at 6% a year, so 0.5% a month. The first instalment is 1,000 + 0.5% · 240,000 = 2,200, and every later one is smaller by 0.5% of 1,000, that is by 5 — the instalment is an arithmetic sequence with r = −5. The sixtieth instalment is 2,200 − 59 · 5 = 1,905, and everything repaid adds up to (2,200 + 1,005) / 2 · 240 = 384,600.
A bacterial culture
A colony doubles every 20 minutes and the starting sample holds 500 cells. The count every 20 minutes is a geometric sequence with q = 2, so three hours later nine doublings have passed and the microbiologist expects 500 · 2⁹ = 256,000 cells. The same formula with q = 0.5 describes radioactive decay: after each half-life, half of the sample is left.
Convert it here: Radioactivity

All formulas

Common difference
$r = a_{n+1} - a_n$
the same for every pair of neighbouring terms
nth term, arithmetic
$a_n = a_1 + (n - 1)r$
the first term plus n − 1 steps
Arithmetic mean
$a_n = \frac{a_{n-1} + a_{n+1}}{2}$
every term is the mean of its neighbours
Sum of n terms, arithmetic
$S_n = \frac{a_1 + a_n}{2} \cdot n$
the mean of the outer terms times how many there are
Common ratio
$q = \frac{a_{n+1}}{a_n}$
the same for every pair of neighbouring terms
nth term, geometric
$a_n = a_1 \cdot q^{n-1}$
the first term times the ratio raised to n − 1
Geometric mean
$a_n^2 = a_{n-1} \cdot a_{n+1}$
a term squared equals the product of its neighbours
Sum of n terms, geometric
$S_n = a_1 \cdot \frac{q^n - 1}{q - 1}, \qquad q \neq 1$
for q = 1 the sum is simply n times a₁

Every sequence in the previous lesson shared one feature: it was not required to be regular at all. The two sequences of this lesson sit at exactly the opposite end — they are built by repeating one and the same action.

Two kinds of regularity

Take a sequence from the previous lesson and look at what happens between neighbouring terms.

If the difference $a_{n+1} - a_n$ is constant, the sequence is called arithmetic, and that constant is the common difference $r$ .
If the ratio $\frac{a_{n+1}}{a_n}$ is constant, the sequence is called geometric, and that constant is the common ratio $q$ .

2, \; 5, \; 8, \; 11, \; 14 \quad (r = 3) \qquad\qquad 2, \; 4, \; 8, \; 16, \; 32 \quad (q = 2)

Both start at two and both look similar at first. After that they part ways for good.

Both sequences start at 2. The arithmetic one adds 3 at every step, the geometric one doubles — after six terms one is at 17, the other at 64.

Adding gives steady growth, multiplying gives growth that keeps accelerating. Half of finance and all of demography rest on that difference.

The arithmetic sequence

Since each step adds $r$ , getting from $a_1$ to $a_n$ takes $n - 1$ steps:

a_n = a_1 + (n - 1)r

That $n - 1$ is the source of the most common slip in the whole topic. Reaching the fourth term takes three steps: $a_4 = a_1 + 3r$ .

In an arithmetic sequence a₁ = −5 and r = 4. Find a₁₂.

The formula also works backwards. Given two terms, the common difference follows from how many steps separate them:

a_7 - a_3 = 4r \qquad \Longrightarrow \qquad r = \frac{a_7 - a_3}{4}

Every term is the mean of its neighbours

The term on the left is $a_n - r$ and the one on the right is $a_n + r$ , so their mean gives $a_n$ back:

a_n = \frac{a_{n-1} + a_{n+1}}{2}

This property characterises the arithmetic sequence: if it holds for every $n$ , the sequence is arithmetic. It is also how a missing term is inserted — the numbers $8$ , $x$ , $20$ form an arithmetic sequence only for $x = 14$ .

The sum of the first n terms

Legend credits the trick to a ten-year-old Gauss, told to add the numbers from $1$ to $100$ . Instead of working through them, he paired them off from the ends:

(1 + 100) + (2 + 99) + (3 + 98) + \ldots = 101 \cdot 50 = 5050

Every pair gives the same total, and there are half as many pairs as numbers. Hence the formula:

S_n = \frac{a_1 + a_n}{2} \cdot n

When the last term is unknown, substitute it from the general formula:

S_n = \frac{2a_1 + (n - 1)r}{2} \cdot n

Find the sum of the first twenty terms of the arithmetic sequence with a₁ = 3 and r = 5.

The geometric sequence

Here each step multiplies by $q$ , and $n - 1$ steps make $q$ raised to $n - 1$ — a power with the index in the exponent:

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

a_1 = 3, \; q = 2 \quad \Longrightarrow \quad 3, \; 6, \; 12, \; 24, \; 48, \; \ldots

Two restrictions follow straight from the definition: $q \neq 0$ , and no term is ever zero. If one were, the ratio of the next term to it would not exist.

The ratio may be negative — the terms then jump above the axis and below it in turn:

A geometric sequence with ratio q = −2. The sign flips with every term, and the distance from zero doubles at every step regardless.

Such a sequence is neither increasing nor decreasing — and is perfectly regular all the same. The regularity of a geometric sequence lives in its ratio, not in its direction.

The geometric mean

The counterpart of the mean property is built on multiplication:

a_n^2 = a_{n-1} \cdot a_{n+1} \qquad \text{so that} \qquad |a_n| = \sqrt{a_{n-1} \cdot a_{n+1}}

The absolute value is not decoration: the product $4 \cdot 9 = 36$ belongs to $4, 6, 9$ and to $4, -6, 9$ alike. That is why an exercise always adds a condition — for instance that the missing term is positive.

The numbers 4, x, 25 form a geometric sequence and x > 0. Find x.

The sum of the first n terms

S_n = a_1 \cdot \frac{q^n - 1}{q - 1}, \qquad q \neq 1

The condition $q \neq 1$ is necessary — otherwise the denominator would be zero. For $q = 1$ the sequence is constant and the sum is simply $n \cdot a_1$ .

Find the sum of the first eight terms of the geometric sequence with a₁ = 3 and q = 2.

Compound interest is a geometric sequence

An account paying $p$ per period multiplies the capital by the same factor $1 + p$ every period. That is precisely the definition of a geometric sequence with ratio $q = 1 + p$ :

a_n = a_1 \cdot (1 + p)^{n-1}

For $10\,000$ at $4\%$ a year the balance after five years is

10\,000 \cdot 1.04^5 = 12\,166.53

Simple interest — $4\%$ charged forever on the original deposit — would give an arithmetic sequence with $r = 400$ and $12\,000$ after five years. The gap of $166.53$ is the interest on interest; over a longer horizon it snowballs, because one sequence adds and the other multiplies. The arithmetic of percentages and the whole machinery behind it are the subject of the separate lesson on compound interest.

Exercises

The set mixes eight questions: for both sequences, a term at a given index, the parameter ( $r$ or $q$ ) read off two given terms, a term inserted between two neighbours, and the sum of the first $n$ terms. Almost every answer is a whole number; at the hardest tier the geometric mean can be an exact value with a root — type it as e.g. 6√2.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

Arithmetic sequence: a₁ = −3, r = −2; a₉ =

Common mistakes

Writing $n$ instead of $n - 1$ — reaching $a_{10}$ takes nine steps, so $a_{10} = a_1 + 9r$ , not $a_1 + 10r$ .
Confusing the difference with the ratio — $r$ is added, $q$ is multiplied; the sequence $2, 6, 18$ has $q = 3$ , not $r = 4$ .
Dividing by zero in the geometric sum — the case $q = 1$ is computed separately, as $n \cdot a_1$ .
Dropping the second sign in the geometric mean — $x^2 = 36$ gives $x = 6$ or $x = -6$ ; the condition in the problem decides which.
Calling a sequence with a negative ratio irregular — $2, -4, 8, -16$ is geometric; it simply is not monotonic.
Adding terms up instead of using the formula — at twenty terms that is a route to an arithmetic slip, not a saving.

Formula card

Topic: Arithmetic and geometric sequences

Common difference
$r = a_{n+1} - a_n$
the same for every pair of neighbouring terms
nth term, arithmetic
$a_n = a_1 + (n - 1)r$
the first term plus n − 1 steps
Arithmetic mean
$a_n = \frac{a_{n-1} + a_{n+1}}{2}$
every term is the mean of its neighbours
Sum of n terms, arithmetic
$S_n = \frac{a_1 + a_n}{2} \cdot n$
the mean of the outer terms times how many there are
Common ratio
$q = \frac{a_{n+1}}{a_n}$
the same for every pair of neighbouring terms
nth term, geometric
$a_n = a_1 \cdot q^{n-1}$
the first term times the ratio raised to n − 1
Geometric mean
$a_n^2 = a_{n-1} \cdot a_{n+1}$
a term squared equals the product of its neighbours
Sum of n terms, geometric
$S_n = a_1 \cdot \frac{q^n - 1}{q - 1}, \qquad q \neq 1$
for q = 1 the sum is simply n times a₁

The arithmetic sequence a(n) = 3n − 1 and the geometric one b(n) = 2 to the power of n on the same axes. The first adds 3 at every step, the second doubles — and after six terms one is at 17 and the other at 64.

A geometric sequence with a negative ratio: q = −2 throws the terms above the axis and below it in turn, while their distance from zero grows all the same.

Frequently asked questions

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min

Arithmetic and geometric sequences

Before you start

Where this is used

All formulas

Two kinds of regularity

The arithmetic sequence

Every term is the mean of its neighbours

The sum of the first n terms

The geometric sequence

The geometric mean

The sum of the first n terms

Compound interest is a geometric sequence

Exercises

Practice

Common mistakes

Formula card

Frequently asked questions

What is an arithmetic sequence?

What is a geometric sequence?

Why does the nth-term formula say n − 1 rather than n?

How do I sum the first n terms of an arithmetic sequence?

How do I sum the first n terms of a geometric sequence?

What does it mean that a term is the mean of its neighbours?

Can the common ratio be negative?

Why is compound interest a geometric sequence?

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