Arithmetic Sequence Calculator: nth Term, Sum, Solve n or d

About this calculator

The arithmetic sequence calculator handles sequences in which every term differs from the previous one by the same amount. Enter the first term $a_1$ , common difference $d$ and number of terms $n$ to get the nth term $a_n = a_1 + (n-1)d$ and the sum of the first $n$ terms $S_n = \dfrac{n(a_1 + a_n)}{2}$ . If the last term is known, it can also solve for the term count $n$ or the common difference $d$ .

The odd numbers 1, 3, 5, 7, …; theater rows that gain 2 seats each row; a savings plan that increases by 50 each month; or training distance reduced by 1.5 km each day are all arithmetic sequences as long as the increment stays constant. It is the simplest sequence model and a useful starting point for understanding accumulation. The end-pairing trick behind the sum formula is famously associated with the young Gauss.

What it does not do: it does not handle geometric sequences, where each term is a fixed multiple of the previous one, as in compound interest. It also does not handle second-order differences or recurrences such as Fibonacci sequences, and it does not list every term. Its primary outputs are the nth term and the total.

Formula and variables

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

Solving from a known last term:

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

Symbol	Meaning	Calculator field	Unit	Range and notes
$a_1$	First term	First term a₁	any	Default 1; may be negative or decimal
$d$	Common difference, $a_{k+1} - a_k$	Difference d	same as $a_1$	Default 2; $d > 0$ increases, $d < 0$ decreases, $d = 0$ is constant
$n$	Number of terms	Term count n	count	Positive integer, $n \ge 1$ ; the panel rejects non-integer or below-1 input (`min: 1, integer: true`). The field has a default of 10, but the formula panel pre-fills only the first two fields, so it opens empty; the public embed calculator applies every prepared default (10)
$a_n$	nth term (last term)	nth term aₙ	same as $a_1$	Output; can be supplied as input in solve mode
$S_n$	Sum of the first $n$ terms	Sum S	same as $a_1$	Output: the total of $n$ terms

The two sum formulas are equivalent. The first multiplies the average of the first and last terms by the number of terms; the second uses only $a_1$ , $d$ and $n$ . The engine evaluates the first form, while the second is useful for hand checking.

How to use this calculator

Open the panelthis page corresponds to “Math tools → Formulas → Arithmetic sequence” (/#/mathtools/formula?calculator=calculator.arithmetic-seq). The formula panel initially pre-fills only “First term a₁” = 1 and “Difference d” = 2; “Term count n” is empty. At that moment the nth term displays −1 and the sum 0 because the engine substitutes an empty $n$ as 0 into $a_1 + (0-1)d$ . Those are not meaningful results; enter $n$ to get the intended calculation. The public embed calculator applies every prepared default from the definition ( $n = 10$ , giving nth term 19 and sum 100).
Find the nth term and sumenter a positive integer in “Term count n”. Both fields marked as computed update together.
Solve for the term countclear “Term count n”, enter the last term in “nth term aₙ”, and keep $a_1$ and $d$ . The term-count field becomes computed. During this mode “Sum” displays 0 because the engine falls back to $n = 0$ ; enter the solved $n$ back into the term-count field and clear $a_n$ to obtain the real sum.
Solve for the common differenceclear “Difference d” and enter $a_1$ , $n$ and $a_n$ . The difference field returns $(a_n - a_1)/(n-1)$ . The sum is temporarily evaluated with $d = 0$ , so it shows $n \cdot a_1$ rather than the real sum. Enter the solved $d$ back into the field to calculate the sum.
Read the resultvalues are shown to at most 8 decimal places with trailing zeros removed. If a solved $n$ is not an integer, such as 16.66666667, the supplied $a_n$ is not a term of that sequence.
Copy the formulathe copy button beside the panel title copies aₙ = a₁ + (n-1)d, S = n(a₁+aₙ)/2 to the clipboard.

Worked examples

All four examples were recalculated with the arithmetic-seq engine compute function. Displayed values use at most 8 decimal places with trailing zeros removed.

Example 1: 3, 7, 11, 15, 19, 23 — six terms

Input: $a_1 = 3$ , $d = 4$ , $n = 6$ .

The sixth term is $a_6 = 3 + (6-1) \times 4 = 3 + 20 = 23$ .
The sum is $S_6 = \dfrac{6 \times (3 + 23)}{2} = \dfrac{6 \times 26}{2} = 78$ .

The UI shows nth term 23 and sum 78. With the alternate form: $6 \times 3 + \dfrac{6 \times 5}{2} \times 4 = 18 + 60 = 78$ . By pairing the ends, $3 + 23 = 7 + 19 = 11 + 15 = 26$ : three pairs of 26 give 78.

Figure 1: Example 1. Left, Gauss pairing makes every first-last pair equal a₁ + aₙ. Right, two copies of the staircase form a rectangle of area n(a₁ + aₙ); one copy is half of it, Sₙ

Example 2: the default input and Gauss's 1 through 100

Input $a_1 = 1$ , $d = 2$ , $n = 10$ : the tenth term is $1 + 9 \times 2 = 19$ and the sum is $10 \times (1 + 19) / 2 = 100$ . The UI shows nth term 19 and sum 100. The first 10 odd numbers sum to $10^2$ . Change $n = 100$ and the UI shows nth term 199 and sum 10000: the first $n$ odd numbers always sum to $n^2$ .

Now enter $a_1 = 1$ , $d = 1$ , $n = 100$ . The UI shows nth term 100 and sum 5050. This is the problem in the well-known story of the young Gauss pairing $1 + 100 = 2 + 99 = \cdots = 50 + 51$ : 50 pairs, each equal to 101.

Example 3: theater seats and a decreasing sequence

Theater: the first row has 20 seats, each later row has 2 more, and there are 15 rows. Enter $a_1 = 20$ , $d = 2$ , $n = 15$ . The last row has $20 + 14 \times 2 = 48$ seats, and the total is $15 \times (20 + 48)/2 = 510$ . The UI shows nth term 48 and sum 510.

Decreasing sequence: suppose inventory starts at 100 kg and 7 kg is shipped each day. Enter $a_1 = 100$ , $d = -7$ , $n = 12$ . Day 12 inventory is $100 - 77 = 23$ , and the sum of daily inventories is $12 \times (100 + 23)/2 = 738$ kg·day, equivalent to an average inventory of 61.5 kg over 12 days. The UI shows nth term 23 and sum 738. The formulas are unchanged when the common difference is negative.

Saving 50 more each month: save 100 in month 1, then 50 more than the previous month for a year. With $a_1 = 100$ , $d = 50$ , $n = 12$ , month 12 is 650 and the annual total is 4500.

Example 4: solving for term count and common difference

Solve for n: which term of 5, 8, 11, … is 50? Enter $a_1 = 5$ , $d = 3$ , $a_n = 50$ and clear n. Then $n = (50 - 5)/3 + 1 = 16$ . The UI shows term count 16 and sum 0 under the solve-mode convention described in step 3. Enter 16 back into the term-count field and clear $a_n$ to get nth term 50 and sum 440.

If the target is 52, $n = (52 - 5)/3 + 1 = 16.66666667$ . That is not an integer, so 52 is not in the sequence: term 16 is 50 and term 17 is 53.

Solve for d: starting at 2 and reaching 30 on term 8, what is the difference? Enter $a_1 = 2$ , $n = 8$ , $a_n = 30$ and clear d. Then $d = (30 - 2)/(8 - 1) = 4$ . The UI shows difference 4 and sum 16 because the engine temporarily evaluates the sum with $d = 0$ , giving $8 \times 2 = 16$ . Enter 4 back into the difference field and clear $a_n$ to get nth term 30 and sum 128.

You can reproduce Example 1 directly in the panel. Change $n$ from 6 to 12, 24 and 48 to see that the nth term grows linearly with $n$ , while the sum grows roughly with $n^2$ : doubling the term count makes the sum about four times as large.

Principle and derivation

nth term: from recurrence to direct formula

The definition is recursive: $a_{k+1} = a_k + d$ . Moving from $a_1$ to $a_n$ requires adding $d$ exactly $n - 1$ times, so $a_n = a_1 + (n-1)d$ . It is $n - 1$ , not $n$ : six bars have only five gaps between them, which is the same off-by-one idea seen in fence-post problems.

Treating $n$ as a variable, $a_n = d \cdot n + (a_1 - d)$ is a straight line with slope $d$ and intercept $a_1 - d$ , the formal “term zero”. An arithmetic sequence is therefore a linear function restricted to positive integer inputs. The tops of the bars in Figure 1 lie on a line. The slope between any two sequence terms is the common difference, and $d = (a_n - a_1)/(n-1)$ is exactly rise divided by run.

Sum: Gauss pairing

Write $S_n$ once in forward order and once in reverse, then add the two rows:

\begin{aligned} S_n &= a_1 + (a_1 + d) + \cdots + (a_n - d) + a_n \\ S_n &= a_n + (a_n - d) + \cdots + (a_1 + d) + a_1 \\ 2S_n &= (a_1 + a_n) + (a_1 + a_n) + \cdots + (a_1 + a_n) = n\,(a_1 + a_n) \end{aligned}

Every column totals $a_1 + a_n$ : when one term is larger by $d$ , the term paired with it is smaller by $d$ . There are $n$ columns, so $S_n = n(a_1 + a_n)/2$ . For an odd number of terms, the middle term pairs with itself; it is exactly $(a_1 + a_n)/2$ , the average of all terms. Figure 1 shows the same argument geometrically: an upright staircase and an inverted copy form a rectangle of size $n \times (a_1 + a_n)$ .

Substituting $a_n = a_1 + (n-1)d$ gives $S_n = n a_1 + \dfrac{n(n-1)}{2} d$ : $n$ copies of $a_1$ plus $0 + 1 + \cdots + (n-1) = n(n-1)/2$ copies of $d$ . This also makes the quadratic dependence on $n$ explicit.

Reading the formula as “average × count”

$S_n = n \cdot \dfrac{a_1 + a_n}{2}$ says that the total equals the average of the first and last terms times the number of terms. In an arithmetic sequence, that endpoint average is also the average of every term because the terms are symmetric around the midpoint. In the inventory example, 738 = 61.5 × 12 is simply average inventory × days. The same structure appears in the trapezoid area formula and in constant-acceleration displacement written as average velocity × time.

Arithmetic versus geometric: simple and compound interest

An arithmetic sequence adds a fixed amount each step; a geometric sequence multiplies by a fixed factor. A principal of 1,000 at 5% simple interest earns a fixed 50 per year, producing 1,050, 1,100, 1,150, … and reaching 1,500 after 10 years. At 5% compound interest the sequence is geometric, $1000 \times 1.05^n$ , reaching 1,628.89 after 10 years. One is linear and the other exponential, so the gap grows with time; see the compound interest calculator for that model. These are illustrative numbers, not financial advice.

Assumptions

Constant difference: every adjacent pair has the same difference. If even one step differs, the formulas no longer describe the whole sequence.
Positive integer term count: the panel rejects non-integer or below-1 input (min: 1, integer: true); only the underlying compute() skips the check, so $n = 2.5$ would produce a meaningless number.
Indexing starts at term 1: $a_1$ is the first term, not term zero. Some textbooks or programs use zero-based indexing, where the corresponding form is $a_n = a_0 + nd$ .
The sum means the first $n$ terms: from $a_1$ through $a_n$ . For terms $m$ through $n$ , use $S_n - S_{m-1}$ .
UI solve-mode convention: blank fields enter the engine as 0, so while solving for $n$ or $d$ , the displayed sum is not the real sum until the solved value is entered back into the main field.

Scope and limitations

Supported: the nth term and first- $n$ sum for real $a_1$ and $d$ with positive integer $n$ ; solving $n$ from $a_n$ when $d \ne 0$ ; solving $d$ when $n \ne 1$ .
Model approximation: real examples such as seats or inventory may only approximately follow a constant difference; the formula reports the model value.
Not supported: geometric sequences, second-order differences such as $1, 3, 6, 10, \ldots$ , an arbitrary partial sum $a_m + \cdots + a_n$ in one operation, or solving $n$ or $d$ directly from $S_n$ .
Not provided: listing every term or directly testing membership. To test whether a value belongs to the sequence, solve for $n$ and check whether it is a positive integer.
Numerics: very large sums may display in scientific notation. JavaScript double precision remains the underlying numerical representation.

Common pitfalls

Writing $a_1 + nd$ : this misses the $-1$ and adds one extra $d$ to every term. Six terms have only five gaps.
Treating a fractional solved n as a valid term number: $n = 16.67$ means the supplied last value is not a sequence term, not that there is a “16.67th term”.
Reading the solve-mode sum as a result: it is a fallback calculation using $n = 0$ or $d = 0$ . Enter the solved value back into its field before reading the real sum.
Using a positive difference for a decreasing sequence: 100, 93, 86, … has $d = -7$ , not 7.
Forgetting the division by 2 in the sum formula: $n(a_1 + a_n)$ is the rectangle made from two staircase copies; one staircase is half of it.
Confusing geometric growth with arithmetic growth: compound interest, population growth and repeated multiplication require a geometric model.

Typical uses

Teaching

Move from a recurrence to a closed form, derive the sum by pairing, observe that the first $n$ odd numbers sum to $n^2$ , and interpret an arithmetic sequence as a linear function sampled at integer points.

Counting and arrangements

Rows of seats that increase regularly, layered stacks whose counts change by a fixed amount, and similar discrete arrangements.

Planning and budgeting

Savings, repayment, inventory or training schedules that change by a fixed amount each period.

Simple interest and straight-line depreciation

Simple interest accumulates by a fixed amount each period; straight-line depreciation reduces book value by a fixed amount, so the nth-period value follows the same arithmetic structure.

FAQ

Why does the nth term show −1 when I first open the panel?

The term count is empty, so the engine substitutes $n = 0$ into $1 + (0-1) \times 2 = -1$ . Enter a term count and the display becomes meaningful; the public embed calculator fills every prepared default ( $n = 10$ ).

How can I test whether a number is in the sequence?

Clear $n$ , enter the candidate as “nth term”, and check whether the solved $n$ is a positive integer. In 5, 8, 11, …, 50 gives $n = 16$ , while 52 gives $n = 16.67$ and is not a term.

How do I sum terms 5 through 12?

Compute $S_{12}$ and $S_4$ and subtract. Alternatively, use term 5 as a new first term with a term count of 8.

If I know the sum, first term and common difference, how do I solve for n?

Solve the quadratic equation $\dfrac{d}{2}n^2 + \left(a_1 - \dfrac d2\right)n - S_n = 0$ and take a positive integer root. This inverse operation is not supported by the calculator.

Why does the sum grow quadratically?

$S_n = n a_1 + \dfrac{n(n-1)}{2}d$ contains an $n^2$ term. Geometrically, the sum is the area under a staircase whose top follows a straight line, so the area grows roughly with the square of the width.

How do I choose between an arithmetic and geometric sequence?

Check whether adjacent terms have a constant difference or a constant ratio. Adding 50 each year is arithmetic; multiplying by 1.05 each year is geometric.

References

CalcX engine source src/data/formulas.ts, arithmetic-seq definition (compute(known, solveFor) produces an / sum and can solve n or d from an); all numerical examples on this page were recalculated with that engine.
Wikipedia, Arithmetic progression （访问日期：2026-09-09） — nth-term and sum formulas and derivations.
Wikipedia, 1 + 2 + 3 + 4 + ⋯ （访问日期：2026-09-09） — triangular numbers $n(n+1)/2$ and the Gauss story.
Brian Hayes, “Gauss's Day of Reckoning”, American Scientist 94(3), 2006 （访问日期：2026-09-09） — historical examination of versions of the Gauss summation story.
Wikipedia, Rhind Mathematical Papyrus （访问日期：2026-09-09） — ancient arithmetic-allocation problems, including problems 40 and 64.
The Nine Chapters on the Mathematical Art, chapters on proportional and transport allocation — examples of arithmetic distribution and summation in the Chinese mathematical tradition.

Privacy

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Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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