What is an arithmetic sequence?

An arithmetic sequence is a sequence in which consecutive terms differ by a constant value called the common difference, denoted d. Starting from a first term a_1, each subsequent term is obtained by adding d to the previous one, producing the pattern a_1, a_1 + d, a_1 + 2d, and so on.

What is the formula for the nth term of an arithmetic sequence?

The general term of an arithmetic sequence is given by a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the position of the term. This explicit formula allows direct computation of any term without needing to know the previous ones.

How do you find the sum of an arithmetic sequence?

The sum of the first n terms of an arithmetic sequence, called an arithmetic series, is S_n = n/2 times (a_1 + a_n). An equivalent form using only the first term and common difference is S_n = n/2 times (2a_1 + (n - 1)d). The formula is derived by pairing terms from opposite ends of the sum.

How do you identify whether a sequence is arithmetic?

Compute the differences between consecutive terms. If every difference equals the same value, the sequence is arithmetic and that value is the common difference d. A single pair of unequal differences is enough to disqualify the sequence from being arithmetic.

What is the arithmetic mean in an arithmetic sequence?

The arithmetic mean of two numbers a and b is (a + b)/2. In an arithmetic sequence, every interior term equals the arithmetic mean of its two neighbors: a_n = (a_(n-1) + a_(n+1))/2. This property follows directly from the constant-difference structure.

Arithmetic Sequences

Definition

Identifying Arithmetic Sequences

Constant Difference

An arithmetic sequence adds the same fixed value to each term to produce the next. That fixed value — the common difference — determines everything: whether the sequence rises or falls, how fast it grows, and what its partial sums look like. Arithmetic sequences are the simplest infinite pattern, and their summation formula is one of the oldest results in mathematics.

Definition

An arithmetic sequence is a sequence in which consecutive terms differ by a constant

d

, called the common difference:

a_{n+1} - a_n = d \quad \text{for all } n

Starting from a first term

a_1

, the sequence unfolds as

a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \ldots

. The general term is:

a_n = a_1 + (n - 1)d

The common difference

d

can be any real number. When

d > 0

, the sequence increases:

3, 7, 11, 15, \ldots

has

d = 4

. When

d < 0

, it decreases:

20, 17, 14, 11, \ldots

has

d = -3

. When

d = 0

, every term equals

a_1

— a constant sequence, which is arithmetic in the trivial sense.

The explicit formula

a_n = a_1 + (n-1)d

has the same structure as a linear function

f(x) = mx + b

: the common difference

d

plays the role of slope, and the sequence's terms lie on a straight line when plotted against their indices.

Identifying Arithmetic Sequences

To determine whether a given sequence is arithmetic, compute the differences between consecutive terms. If every difference equals the same value, the sequence is arithmetic and that value is

d

.

The sequence

5, 12, 19, 26, 33

has differences

7, 7, 7, 7

. All equal, so it is arithmetic with

d = 7

.

The sequence

1, 4, 9, 16, 25

has differences

3, 5, 7, 9

. These are not constant, so the sequence is not arithmetic. (It is the sequence of perfect square numbers, which grows quadratically rather than linearly.)

The test applies to any number of terms: compute all consecutive differences and check for uniformity. A single pair of unequal differences is enough to disqualify. Conversely, if only a few terms are given, the test cannot confirm an arithmetic pattern with certainty — the sequence

2, 4, 6

could continue as

8, 10, \ldots

(arithmetic) or as

10, 16, \ldots

(something else). The rule generating the sequence, not just a finite sample, is what determines its type.

Finding Terms

The explicit formula

a_n = a_1 + (n-1)d

contains four quantities:

a_n

a_1

n

, and

d

. Given any three, the fourth can be found by solving the equation.

Finding a specific term: if

a_1 = 4

and

d = 6

, then

a_{20} = 4 + 19 \cdot 6 = 118

.

Finding the common difference: if

a_1 = 3

and

a_8 = 24

, then

24 = 3 + 7d

, so

d = 3

.

Finding the first term: if

a_{12} = 50

and

d = 4

, then

50 = a_1 + 11 \cdot 4

, so

a_1 = 6

.

Finding the position of a term: if

a_1 = 7

d = 5

, and some term equals

52

, then

52 = 7 + (n-1) \cdot 5

, so

n - 1 = 9

and

n = 10

. The value

52

is the

10

th term. If the equation yields a non-integer

n

, the value does not appear in the sequence.

When two non-consecutive terms are given — say

a_3 = 11

and

a_9 = 35

— write both in terms of

a_1

and

d

a_1 + 2d = 11

and

a_1 + 8d = 35

. Subtracting gives

6d = 24

, so

d = 4

, and back-substituting gives

a_1 = 3

Recursive Formula

The recursive definition of an arithmetic sequence specifies a starting value and a rule for getting from one term to the next:

a_1 = c, \quad a_n = a_{n-1} + d

Each term is built from the immediately preceding term by adding

d

. This formulation makes the constant-difference structure explicit but requires computing all earlier terms to reach a later one.

Converting from recursive to explicit: if

a_1 = c

and

a_n = a_{n-1} + d

, then after applying the rule

n - 1

times,

a_n = c + (n-1)d

. The explicit formula is a direct consequence of adding

d

repeatedly.

Converting from explicit to recursive: given

a_n = a_1 + (n-1)d

, observe that

a_n - a_{n-1} = d

, which gives the recursive rule

a_n = a_{n-1} + d

with initial value

a_1

.

Both representations define the same sequence. The explicit form is better for computing isolated terms; the recursive form is better for understanding the building process and connects naturally to the idea of sequences defined by recurrence.

Arithmetic Series

The sum of the first

n

terms of an arithmetic sequence is called an arithmetic series, denoted

S_n

S_n = a_1 + a_2 + a_3 + \cdots + a_n

The closed-form formula is:

S_n = \frac{n}{2}(a_1 + a_n)

The derivation is straightforward. Write the sum forwards and backwards:

S_n = a_1 + (a_1 + d) + (a_1 + 2d) + \cdots + a_n

S_n = a_n + (a_n - d) + (a_n - 2d) + \cdots + a_1

Adding these two expressions term by term, every pair sums to

a_1 + a_n

, and there are

n

such pairs:

2S_n = n(a_1 + a_n)

Dividing by

2

gives the result. An equivalent form replaces

a_n

with

a_1 + (n-1)d

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

This version is useful when

a_n

is not known directly.

The sum of the first

100

natural numbers, famously attributed to the young Gauss:

a_1 = 1

a_{100} = 100

, so

S_{100} = \frac{100}{2}(1 + 100) = 5050

Arithmetic Mean

The arithmetic mean of two numbers

a

and

b

is:

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

In an arithmetic sequence, every term (except the first and last) is the arithmetic mean of its two neighbors:

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

This follows directly from the constant-difference property: since

a_n - a_{n-1} = a_{n+1} - a_n = d

, adding the neighbors gives

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

.

Inserting arithmetic means between two values is the reverse problem: given endpoints

a

and

b

, place

k

terms between them so that the resulting

k + 2

values form an arithmetic sequence. The common difference must be

d = \frac{b - a}{k + 1}

, and the inserted terms are

a + d, a + 2d, \ldots, a + kd

.

For example, inserting

3

arithmetic means between

5

and

25

requires

d = \frac{25 - 5}{4} = 5

, producing the sequence

5, 10, 15, 20, 25