What is the formula for an arithmetic sequence?

The n-th term formula is a sub n equals a sub 1 plus the quantity n minus 1, times d, where a sub 1 is the first term and d is the common difference. The sum of the first n terms is given by S sub n equals n divided by 2, times the quantity 2 a sub 1 plus the quantity n minus 1, times d, or equivalently S sub n equals n times the quantity a sub 1 plus a sub n, divided by 2.

How do you find the common difference?

Subtract any term from the next one. If a sub k and a sub k plus 1 are consecutive terms, then d equals a sub k plus 1 minus a sub k. If you know two non-consecutive terms a sub m and a sub n, then d equals the quantity a sub n minus a sub m, divided by the quantity n minus m.

How do you find the sum of an arithmetic sequence?

The sum of the first n terms equals n divided by 2 times the sum of the first and last terms: S sub n equals n divided by 2, times the quantity a sub 1 plus a sub n. Equivalently, in terms of just a sub 1 and d, S sub n equals n divided by 2, times the quantity 2 a sub 1 plus the quantity n minus 1, times d. The formula was famously rediscovered by Gauss as a young student.

How is an arithmetic sequence different from a geometric sequence?

An arithmetic sequence adds a constant common difference d at each step; a geometric sequence multiplies by a constant common ratio r at each step. Arithmetic grows linearly: a sub n equals a sub 1 plus the quantity n minus 1, times d. Geometric grows exponentially: a sub n equals a sub 1 times r to the power of n minus 1. Both are foundational sequence types in algebra.

Arithmetic Sequence Calculator — Solve a₁, d, n, aₙ, Sₙ

Q: What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where each term differs from the previous by a constant called the common difference. Starting from a first term a sub 1 and adding the common difference d each step, the n-th term is given by a sub n equals a sub 1 plus the quantity n minus 1, times d. Examples include 2, 5, 8, 11, 14 (with d equals 3) and 10, 7, 4, 1, minus 2 (with d equals negative 3).

Setting the Parameters

DemoTwo new parameter pairs, then the opening pair again

Step 0 of 3

The card opens with the Sequence drop-down set to Arithmetic sequence; the other entry switches the card to the Geometric sequence, which has its own page, Geometric sequence calculator. Under it the Parameters row holds the two numbers that fix the sequence, First term a₁ and Common difference d, opening with

a_1 = 2

and

d = 3

. Type new values and click Apply or press Enter; every tab recomputes at once with the rule

a_n = a_1 + (n - 1) \cdot d

. With

a_1 = 5

and

d = 2

the first eight terms run

5, 7, \ldots, 19

with the sum

96

; with

a_1 = 10

and

d = -3

they fall from

10

to

-11

and the sum is

-4

.

Both boxes take any decimal, negative values included; a blank or non-numeric box is refused with "Enter a number in every parameter box." and the tabs keep their last values. Which two numbers define the sequence is explained in what is an arithmetic sequence.

Listing the First N Terms

DemoThree counts in List first N

Step 0 of 3

The explorer opens on the List first N tab, one of five tabs across the top of the card; the small ? beside each tab shows what the mode does. Type how many terms you want in the Show first box and click Update or press Enter. Three cards answer at once: Showing gives the count, Largest the last term with its index, and Sum the total of the listed terms,

S_N = \frac{N}{2}(a_1 + a_N)

. Under the cards every term is laid out as a tile with its index above its value.

With the opening parameters the box accepts

1

to

500

:

8

gives

a_8 = 23

with the sum

100

,

10

gives

a_{10} = 29

with the sum

155

, and

600

is capped at

500

terms, ending at

a_{500} = 1,499

with the sum

375,250

; a single term gives

a_1 = 2

. Large values are printed with thousands separators. A count of 0, a decimal or an empty box is refused with "Enter a positive integer." There is no Reset button: type a new count and click Update again. The panel on the right, List first N, states the rule the tiles come from, and its walkthrough repeats the last term and the sum with your own

N

. The rule is explained in the n-th term formula and the sum in the sum formula and gauss's trick.

Listing a Range of Indexes

DemoThe range tab, a new range, two bad ranges

Step 0 of 4

Click the List a – b tab to list the terms between two positions instead of starting from

1

. Type the From index and the To index and click List range or press Enter. The cards give the Count

b - a + 1

, the Smallest term

a_a

and the Largest term

a_b

, and the tiles below list every term in between with its index. The tab opens with

5

to

12

, 8 terms from

a_{5} = 14

to

a_{12} = 35

; indexes

10

to

20

give 11 terms from

29

to

59

.

Both boxes must hold positive integers, the first no larger than the second, and the range may hold at most

500

terms; otherwise a red box explains: "Enter two positive integer indexes", "From-index must be less than or equal to to-index", or "Range too wide — cap is 500 terms". The panel on the right, List a – b, explains the mode, and its walkthrough names the smallest and largest term of your range. The rule each tile uses is explained in the n-th term formula.

Testing a Number

DemoThe test tab, a non-member, a member, a decimal

Step 0 of 4

Click the Test a number tab to ask whether a given number is a term of the sequence. Type it in the Test number box and click Check or press Enter. A green box with a tick answers yes and names the index, as for the opening value

47

, which is

a_{16}

; a red box with a cross answers no, as for

49

, and the walkthrough on the right then shows the two nearest terms,

a_{16} = 47

and

a_{17} = 50

. A large value is handled the same way:

2,999

is

a_{1000}

.

The box takes any number, decimals and negatives included, since the parameters may be decimals too; an empty box is refused with "Enter a number". The test is exact: the explorer computes

k = (m - a_1) / d

and accepts

m

only when

k

is a whole number that is not negative; the index is then

k + 1

. The panel on the right, Test a number, states this rule, and its walkthrough carries out the arithmetic for your number. The formula behind it is in the n-th term formula.

Looking Up a Term by Index

DemoThe lookup tab, two indexes, n = 0

Step 0 of 4

Click the Lookup by index tab to compute one term directly. Type the position in the Index n box and click Get term or press Enter. Two cards answer: Index repeats

n

and Value gives

a_n

with its index as a subscript. The tab opens with

n = 20

, which gives

a_{20} = 59

;

n = 100

gives

299

and

n = 1000

gives

2,999

. The box takes positive integers only;

0

is refused with "Enter a positive integer index".

The panel on the right has two parts in every mode. How it works names the rule the mode uses, here

a_n = a_1 + (n - 1) \cdot d

, and links to the section of this page that explains it; Walkthrough applies the rule to the numbers you typed, so for the lookup it shows the arithmetic with your

n

and the current parameters filled in. Both change as soon as you switch tabs or click a button. The rule and where it comes from are in the n-th term formula.

Using the Solver Tab

DemoThe solver tab: a term, an unknown moved, a sum

Step 0 of 4

Click the Solver tab, the fifth tab, to work the formulas backwards. Two buttons choose the relation: Find term (aₙ) uses

a_n = a_1 + (n - 1) \cdot d

and Sum Sₙ uses

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

. Each relation has four boxes; the one marked solve with a ? is the unknown and the other three must be filled. The unknown opens as

a_n

or

S_n

; click the solve radio beside another box to make that the unknown. Click Solve or press Enter and the answer appears under the grid, while the walkthrough on the right shows the substitution.

With

a_1 = 2

,

d = 3

and

n = 20

the term is

a_{20} = 59

; with

a_n = 299

and the unknown moved to

n

the solver returns

n = 100

; with

n = 10

the sum is

S_{10} = 155

. Switching the relation clears the boxes, and clicking Solve with a box empty gives the message "Fill all fields except the one being solved for". Solving

S_n

for

n

solves a quadratic and reports "No positive solution for n" when none fits. The solver boxes are independent of the Parameters row. Every relation is explained in using the solver — five quantities, four relations.

What is an Arithmetic Sequence?

An arithmetic sequence (also called an arithmetic progression, AP) is a sequence where each term is obtained by adding a constant to the previous term. The constant is called the common difference, denoted

d

.

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

Examples.

•

a_1 = 2

,

d = 3

: gives

2, 5, 8, 11, 14, 17, 20, 23, \ldots

•

a_1 = 10

,

d = -3

: gives

10, 7, 4, 1, -2, -5, -8, \ldots

•

a_1 = 0

,

d = 1

: gives

0, 1, 2, 3, 4, \ldots

— the natural numbers.
•

a_1 = 1

,

d = 2

: gives

1, 3, 5, 7, 9, \ldots

— the odd numbers.
•

a_1 = 100

,

d = 0

: gives

100, 100, 100, \ldots

— the constant sequence.

Two key formulas.

• The

n

-th term:

a_n = a_1 + (n - 1) d

.
• The sum of the first

n

terms:

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

.

Parametric nature. Unlike triangular or square numbers (where the sequence is fixed), an arithmetic sequence depends on two parameters:

a_1

and

d

. Different choices give different sequences. This explorer lets you set both parameters and explore the resulting sequence.

For deeper coverage see arithmetic sequence, arithmetic progression, and series.

The definition and the notation are explained in Arithmetic sequences: definition and Sequence notation; the overview page compares the main types in Arithmetic sequences.

Finding terms works the formula through more examples, and Recursive formula gives the step-by-step form

a_n = a_{n-1} + d

.

Arithmetic series proves the sum formula and applies it; Explicit and recursive definitions places sums next to the term rules.

Each relation the solver uses is derived in Finding terms and Arithmetic series.

The mean property is the subject of Arithmetic mean; the recurrence is in Recursive formula.

The n-th Term Formula

Derivation. Starting from

a_1

, the next term is

a_1 + d

, then

a_1 + 2d

, then

a_1 + 3d

. After

n - 1

additions, the

n

-th term is

a_n = a_1 + (n - 1) d

Inverting the formula. Given

a_n

,

a_1

, and

d

, solve for

n

:

n = \frac{a_n - a_1}{d} + 1

This works only when

d \neq 0

; if

d = 0

, every term equals

a_1

and

n

is undetermined.

Examples.

• Sequence

5, 8, 11, 14, \ldots

:

a_1 = 5

,

d = 3

. The 20th term is

a_{20} = 5 + 19 \cdot 3 = 62

.
• Sequence

100, 97, 94, \ldots

:

a_1 = 100

,

d = -3

. The 10th term is

a_{10} = 100 + 9 \cdot (-3) = 73

.
• Membership test: is 47 in the sequence

2, 5, 8, 11, \ldots

? Solve

47 = 2 + (n-1) \cdot 3 \implies n = 16

. Since

n

is a positive integer, yes —

47 = a_{16}

.
• Non-membership: is 50 in the sequence

2, 5, 8, 11, \ldots

? Solve

50 = 2 + (n-1) \cdot 3 \implies n = 17

. Wait,

17

is an integer — so

50

IS in the sequence. Let me try

51

:

51 = 2 + (n-1) \cdot 3 \implies n = 17.33

, not an integer. So

51

is not in the sequence.

Two-parameter recovery. Given any two terms

a_m

and

a_n

with

m \neq n

, both

a_1

and

d

can be recovered:

d = (a_n - a_m)/(n - m)

and

a_1 = a_m - (m - 1) d

.

The Sum Formula and Gauss's Trick

The sum formula. The sum of the first

n

terms of an arithmetic sequence is

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

Gauss's trick. Write the sum forward and backward:

\begin{aligned} S_n &= a_1 + a_2 + a_3 + \cdots + a_n \\ S_n &= a_n + a_{n-1} + a_{n-2} + \cdots + a_1 \end{aligned}

Adding column by column: each pair

(a_k + a_{n - k + 1})

has the same value, because in an arithmetic sequence the sum of equally-spaced-from-the-ends terms is constant. Specifically,

a_k + a_{n - k + 1} = a_1 + a_n

for every

k

. Summing

n

such pairs:

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

The Gauss schoolboy story. Legend says young Carl Friedrich Gauss was asked to sum

1 + 2 + \cdots + 100

. He pictured the trick instantly: pair up

1 + 100, 2 + 99, \ldots, 50 + 51

, getting fifty pairs of 101, total

5050

. Whether the story is literal or apocryphal, the technique works for any arithmetic sequence.

Average-of-endpoints interpretation.

S_n = n \cdot \overline{a}

where

\overline{a} = (a_1 + a_n)/2

is the average of the first and last terms — which is also the arithmetic mean of the entire sequence (a special property of arithmetic sequences).

Solving for $n$ . Given

S_n

,

a_1

, and

d

, the equation

S_n = (n/2)(2 a_1 + (n - 1) d)

is quadratic in

n

. Solving:

d n^2 + (2 a_1 - d) n - 2 S_n = 0

The quadratic formula gives

n

. There can be two positive solutions; the smaller is usually the intended one.

Using the Solver — Five Quantities, Four Relations

The arithmetic-sequence explorer includes a solver mode that exploits the relationships between the five quantities

a_1, d, n, a_n, S_n

.

Two solver sub-modes.

• Find term ( $a_n$ ) — uses the relation

a_n = a_1 + (n - 1) d

. Fill any three of

\{a_1, d, n, a_n\}

and the solver computes the fourth.
• Sum $S_n$ — uses the relation

S_n = (n/2)(2 a_1 + (n - 1) d)

. Fill any three of

\{a_1, d, n, S_n\}

and the solver computes the fourth.

Solving for $a_n$ . Direct substitution:

a_n = a_1 + (n - 1) d

.

Solving for $a_1$ . Rearrange:

a_1 = a_n - (n - 1) d

(term sub-mode) or

a_1 = S_n/n - (n - 1) d/2

(sum sub-mode).

Solving for $d$ .

d = (a_n - a_1)/(n - 1)

(term sub-mode) or

d = 2(S_n - n a_1)/(n(n - 1))

(sum sub-mode). Both undefined when

n = 1

.

Solving for $n$ .

n = (a_n - a_1)/d + 1

(term sub-mode, undefined when

d = 0

). For sum sub-mode, solve the quadratic

d n^2 + (2 a_1 - d) n - 2 S_n = 0

; there may be two positive solutions, and the smaller positive integer is usually the answer.

Edge cases the solver handles.

•

d = 0

: the sequence is constant;

n

is undetermined for the term sub-mode (any

n

gives the same value) and reduces to

S_n / a_1

for the sum sub-mode.
•

n = 1

:

d

cannot be recovered from a single term.
• Negative or fractional

n

in the quadratic: rejected (no real positive solution).

Practical use. The solver is helpful for word problems: given partial information about an arithmetic sequence, recover the missing quantity. Example: "The 5th term of an arithmetic sequence is 17 and the 10th term is 32. Find the 50th term." Solve for

a_1

and

d

from the two equations, then compute

a_{50}

.

Properties and Identities

• Recurrence:

a_n = a_{n-1} + d

with given

a_1

. The defining recurrence of any arithmetic sequence.
• Term in terms of any other term:

a_n = a_m + (n - m) d

. The difference of two terms equals the index gap times

d

.
• Symmetry of sums:

a_k + a_{n - k + 1} = a_1 + a_n

for any

k

from 1 to

n

. The pairing property behind Gauss's trick.
• Arithmetic mean is the middle term: the average of

n

consecutive terms of an arithmetic sequence equals the middle term (if

n

is odd) or the average of the two middle terms (if

n

is even). Always equals

(a_1 + a_n)/2

.
• Sum identity:

S_n = n \overline{a}

, the count times the average.
• Sum of squares: not a simple closed form unless reformulated.

\sum a_k^2

involves

\sum k^2

and is degree 3 in

n

.
• Difference of sums:

S_{n} - S_{n-1} = a_n

, the partial sums recover the original sequence.
• Common difference from sum:

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

(the second difference of partial sums is the common difference).

Common Applications

• Linear pricing and depreciation. A car loses a fixed amount per year — its value follows an arithmetic sequence. Pricing schedules, simple interest, and linear cost models all use arithmetic sequences.
• Counting in lattices. Lattice points along a line with rational slope form an arithmetic sequence in one of the coordinates.
• Physics: uniform motion and constant-acceleration sums. Position under constant velocity follows an arithmetic sequence at equal time intervals. Velocity under constant acceleration does the same.
• Calendars and timing. Every 4th year is a leap year (modulo century rules). Recurring schedules form arithmetic progressions.
• Tournament round counts. Round-robin and elimination tournaments often have round counts following arithmetic patterns.
• Stack and pile counts. Arrangements of objects with constant-difference layer counts (rare in nature, common in puzzles).
• Simple interest. Account balance grows by

P r

each year, an arithmetic sequence with

a_1 = P

and

d = P r

. (Compound interest is geometric.)
• Solving word problems. Many classical algebra problems reduce to arithmetic sequences: "Anna's salary increases by \

3,000 each year. After 10 years she earns \

70,000. What was her starting salary?" — find

a_1

given

a_{11} = 70000

and

d = 3000

.

Identifying arithmetic sequences shows how to spot a constant difference in data; Arithmetic sequences at a glance collects the formulas used here.

Common Mistakes

• Off-by-one in the $n$ -th term formula.

a_n = a_1 + (n - 1) d

, not

a_1 + n d

. The 1st term is

a_1

(zero

d

's added), the 2nd term is

a_1 + d

(one

d

added). Many wrong answers come from forgetting the

-1

.

• Wrong sum formula.

S_n = n(a_1 + a_n)/2

, or equivalently

S_n = n(2 a_1 + (n - 1) d)/2

. Common errors: using

n a_1

(missing the

a_n

contribution), or

(a_1 + a_n)/2

(missing the multiplication by

n

).

• Negative common difference confusion. When

d < 0

, the sequence decreases. All formulas still work; just be careful with signs. The sum is still

n(a_1 + a_n)/2

, but

a_n < a_1

now.

• Confusing arithmetic and geometric. Arithmetic adds

d

; geometric multiplies by

r

. "Doubling each step" is geometric (

r = 2

), not arithmetic.

• Index conventions. Some sources start at

a_0

instead of

a_1

, giving

a_n = a_0 + n d

. This explorer uses

a_1

as the first term.

• Trying to use the formula for non-integer $n$ .

a_n = a_1 + (n - 1) d

is mathematically defined for any real

n

, but only integer

n

corresponds to actual sequence positions.

• Forgetting the second positive root in sum-for- $n$ quadratic. The quadratic

d n^2 + (2 a_1 - d) n - 2 S_n = 0

can have two positive solutions. The smaller (usually integer) one is the intended answer; the other might be an extraneous positive root or a valid second possibility for specific

S_n

values.

• Treating $d = 0$ as a special exception incorrectly. When

d = 0

, the sequence is constant,

a_n = a_1

for all

n

, and

S_n = n a_1

. Many formulas still work; just verify before applying.

The one-page summary Arithmetic sequences at a glance is a quick check against these slips; the definition is in Arithmetic sequences: definition.

Related Sequences and Concepts

Geometric Sequence — multiplies by a constant ratio:

a_n = a_1 r^{n-1}

. Arithmetic's exponential counterpart.

Harmonic Sequence — reciprocals of an arithmetic sequence:

1, 1/2, 1/3, 1/4, \ldots

. Not arithmetic itself but built from one.

Arithmetic Series — the sum of an arithmetic sequence.

S_n = n(a_1 + a_n)/2

.

Triangular Numbers — the sums of the arithmetic sequence

1, 2, 3, \ldots

.

T_n = n(n+1)/2

is the special case of the arithmetic-series formula with

a_1 = 1

,

d = 1

.

Square Numbers — the sums of the arithmetic sequence

1, 3, 5, 7, \ldots

.

S_n = n^2 =

sum of first

n

odd numbers.

Pentagonal Numbers — sums of the arithmetic sequence

1, 4, 7, 10, \ldots

(

d = 3

).

Polygonal Numbers — in general, sums of arithmetic sequences with

d = r - 2

for the

r

-gonal family.

Linear Function —

f(n) = a_1 + (n - 1) d

is a linear function in

n

. Arithmetic sequences are discrete linear functions.

Arithmetic Mean — the average. For an arithmetic sequence, the mean of any consecutive run equals the middle term.

Common Difference — the parameter

d

. Defines the sequence's slope.

Arithmetic Progression — another name for arithmetic sequence. Used interchangeably.

Linear Recurrence — the general class. Arithmetic sequences satisfy the first-order linear recurrence

a_n = a_{n-1} + d

.

Gauss Summation — the classical identity

1 + 2 + \cdots + n = n(n+1)/2

, a special case of the arithmetic-series formula.

Simple Interest — financial application; account balance grows arithmetically with annual interest.

Geometric sequences, Harmonic sequences and Triangular numbers are introduced on the sequences overview page; the geometric case has its own lesson in Geometric sequences: definition.