Number of terms in an arithmetic sequence
From 3 to 27 in steps of 4 there are 7 terms: enter the first term, the last term and the common difference, and get the count n with both ends included. Decreasing sequences work too. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- First
- 3
- Last
- 27
- Difference
- 4
- N
- 7
- Sequence hint
- 3, 7, …, 27 (7 terms)
An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a fixed common difference. Students, tutors, and automation scripts often know the first value, the last value, and that common difference, and need a reliable answer to a single question: how many terms does the sequence contain? This number of terms in an arithmetic sequence calculator applies the closed-form identity n equals open parenthesis last minus first close parenthesis divided by d, plus one, and returns that integer term count when the last term truly lies on the progression.
What the term-count formula means and when it applies
In an arithmetic sequence the first term is often written a₁ or a, the common difference is d, and the n-th term is aₙ equals a plus open parenthesis n minus one close parenthesis times d. If you already know the last term aₙ, the first term a, and d, you can rearrange that identity to solve for the unknown index n: subtract a from aₙ, divide by d, then add one. The result n is the number of terms from a through aₙ inclusive.
When d is non-zero, n must still be a positive integer. When last equals first, the formula yields n equals one, which is correct for a single-term sequence. Decreasing sequences with negative d are fully supported: for example first equals ten, last equals two, and d equals negative two yields n equals five for the list ten, eight, six, four, two.
What you can do with it
Classroom and homework checks
Let students enter first term, last term, and common difference to verify the inclusive term count n against the rearranged aₙ formula.
Exam and LMS auto-grading
Mark arithmetic-sequence items with a deterministic oracle that rejects zero d and off-lattice last terms instead of inventing fractional n.
Notebooks and stepped ranges
Count how many values sit from start to end with a fixed step in measurement ladders, billing periods, or machine cycles before summing elsewhere.
FAQ
How do I find the number of terms in an arithmetic sequence?
Send finite numbers first (a₁), last (aₙ), and difference (d ≠ 0).
What if the last term is not on the sequence?
If (last − first) / d is not an integer, last is not reachable from first with that step.
Does the calculator support decreasing sequences?
Yes. Use a negative difference when each term is smaller than the previous one.
What is the formula for the number of terms in an arithmetic sequence?
Subtract the first term from the last, divide by the common difference and add one: n = (last − first) / d + 1. With 3, 27 and a difference of 4, that gives (27 − 3) / 4 + 1 = 7 terms.
Why can't the common difference be zero?
With a difference of zero every term is the same, so the list never reaches a different last value and no term count can be defined. Enter a positive or a negative difference instead.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/arithmetic-number-of-terms \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"first":3,"last":27,"difference":4}'const res = await fetch("https://api.kit.forhosting.com/algebra/arithmetic-number-of-terms", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"first": 3,
"last": 27,
"difference": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/arithmetic-number-of-terms",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"first": 3,
"last": 27,
"difference": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/arithmetic-number-of-terms", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"first":3,"last":27,"difference":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"first":3,"last":27,"difference":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/arithmetic-number-of-terms", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"first": 3,
"last": 27,
"difference": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.arithmetic_number_of_terms",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
When the difference is zero, or when the last value is not reachable by steps of d from the first, the task fails with invalid_input instead of inventing a fractional or negative n.
What the term-count formula means and when it applies
That count is only meaningful when d is not zero: a zero common difference produces a constant list that either never reaches a different last value or never ends, so this calculator rejects d equals zero with a clear invalid_input message. Fractional first terms and fractional steps work the same way as long as the ratio that defines n is integral within a tiny floating-point tolerance used only to absorb binary rounding, not to invent off-lattice endpoints.
Input contract, response shape, and error behaviour
A zero difference is rejected because a finite term count from first to last cannot be defined for a constant common difference of zero. If last is not reachable from first in whole steps of difference, the error message states that n would not be an integer and quotes the raw non-integer value for debugging. Term counts below one or above one billion are also rejected so clients never receive a silently absurd n from extreme micro-differences. Example: first equals three, last equals twenty-seven, difference equals four gives n equals seven, because three plus six times four is twenty-seven. POST a JSON body with three required finite numbers: first (aliases a, a1, first_term), the starting term of the progression; last (aliases an, last_term), the ending term you want to count up to; and difference (aliases d, common_difference), the non-zero common step between consecutive terms. Each field may arrive as a JSON number or as a numeric string with an optional sign, decimal point, or scientific exponent. Missing fields, empty strings, non-finite values, and non-numeric types fail with invalid_input naming the field. On success the body echoes first, last, and difference rounded to twelve decimal places for stable cross-runtime goldens; returns n as a positive integer; restates the formula string n equals open parenthesis last minus first close parenthesis slash d plus one; and includes a short sequence_hint such as three, seven, ellipsis, twenty-seven for classroom display.
Where a number-of-terms calculator earns its keep
Classroom platforms and tutoring tools use a number of terms in an arithmetic sequence calculator so learners can check inclusive counts without off-by-one mistakes when they rearrange aₙ equals a plus open parenthesis n minus one close parenthesis d by hand. Exam generators and LMS auto-graders need a deterministic oracle that marks wrong both a zero difference and an off-lattice last term, instead of silently rounding fractional n and awarding partial credit. Keep requests small, pin n and the echoed coefficients in tests, and you get reproducible term counts for homework, LMS plugins, and edge workers without standing up a full computer-algebra system. Spreadsheet and notebook workflows often know the endpoints of a billing period, a machine cycle, or a measurement ladder and the fixed step size, and only need n before computing a partial sum with a separate series tool. Pair this endpoint with an nth-term or series-sum calculator when you already know n and need the value or the sum; this capability answers only the count question given first, last, and d.
Technical questions
What happens if the common difference is zero?
The task fails with invalid_input. A zero difference cannot define a finite term count from first to last, so the request is not billed.
What does the number-of-terms calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.