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

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 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

.

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.

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.