Arithmetic Sequence Calculator Online Free

Some lists of numbers wander. Others march. Three, seven, eleven, fifteen, nineteen: each step adds the same amount, four, and once you notice that, you can predict the whole list without writing it out. A row of seats in a theatre, a savings plan that adds the same deposit every month, a runner adding the same distance to a weekly long run: all of these are arithmetic sequences, and the arithmetic sequence calculator at the top of this page finds any term in such a sequence and adds up any number of its terms, from just the first term and the step size.

That is more useful than it sounds, because the interesting questions about a sequence are rarely about the first few terms, which you can list by hand. They are about the fiftieth term, or the hundredth, or the total after a year of monthly deposits. Listing a hundred terms by hand is slow and invites exactly the kind of small addition slip that ruins the rest of the list. The formulas behind this calculator jump straight to the term you want and to the total, and this article explains those formulas so you can trust the jump and check it.

What It Is

An arithmetic sequence, also called an arithmetic progression, is a list of numbers in which each term after the first is found by adding a fixed number, called the common difference, to the previous term. The first term is usually written a1, and the common difference is written d. If a1 is 3 and d is 4, the sequence is 3, 7, 11, 15, 19, 23, and so on, each term four more than the last. If d is negative, the sequence decreases: with a1 20 and d negative 3, the sequence is 20, 17, 14, 11, 8, and so on. If d is zero, every term is the same, and the sequence is constant. The defining feature is always the same: one fixed step, repeated. Two questions dominate work with arithmetic sequences. The first is the nth term question: what is the value of the term in position n? The answer is the general term formula, an = a1 + (n − 1) × d. The reasoning is simple once seen. To get from the first term to the second, you add d once. To get to the third, you add it twice. To get to the nth term, you add it n − 1 times, because there are n − 1 steps between the first term and the nth. The formula is just that counting, written compactly. The second question is the sum question: what do the first n terms add up to? The answer is the arithmetic series sum formula, Sn = n/2 × (2a1 + (n − 1) × d), which is the same as n/2 × (a1 + an), the number of terms divided by two, times the sum of the first and last terms. The calculator gives both answers together, because in practice you usually want both: where the sequence reaches, and how much it totals along the way. The distinction between a sequence and a series is worth making, because textbooks use the two words for different things and mixing them up causes confusion. A sequence is the list itself: 3, 7, 11, 15. A series is the sum of a sequence's terms: 3 + 7 + 11 + 15 = 36. This tool handles both sides of that pair. It lists the terms so you can see the sequence, and it computes the sum so you can use the series. When people search for an arithmetic sequence calculator, they usually want both, and giving only one would leave half the job undone.

How to Use the Calculator

Using the tool takes less than a minute. Follow these steps:

1. Enter the first term a1. This can be any number, positive, negative, or zero, whole or decimal.
2. Enter the common difference d, the fixed amount added at each step. Use a negative number for a decreasing sequence and zero for a constant one.
3. Enter n, the number of terms to count. This must be a whole number of 1 or more.
4. Press "Calculate". The tool shows the nth term with its working, the sum of the first n terms with its working, and the sequence itself.
5. Read the working, not just the answers. The formulas are shown with your numbers in place, so you can follow each step and spot any input slip before you rely on the result.

A common input error is worth warning about in advance. The nth term formula uses n − 1, not n, and forgetting that minus one is the single most frequent mistake in sequence problems. If you want the tenth term of the sequence starting at 3 with difference 4, you add the difference nine times, not ten, because the first term already counts as term one. The calculator handles this for you, but if you check its working by hand and get 43 instead of 39, you have almost certainly added the difference ten times. Count the steps between terms, not the terms themselves, and the formula becomes obvious.

Worked Examples

Example 1: a1 = 3, d = 4, n = 10, the live test. This is the example used to test the tool. The sequence begins 3, 7, 11, 15, 19, 23, 27, 31, 35, 39. For the tenth term, the formula gives a10 = 3 + (10 − 1) × 4 = 3 + 36 = 39, which matches the last term of the list. For the sum, Sn = 10/2 × (2 × 3 + (10 − 1) × 4) = 5 × (6 + 36) = 5 × 42 = 210. You can check that by adding the list by hand: 3 + 7 is 10, plus 11 is 21, plus 15 is 36, plus 19 is 55, plus 23 is 78, plus 27 is 105, plus 31 is 136, plus 35 is 171, plus 39 is 210. The formula and the long addition agree, as they must, and the formula gets there in one line where the addition takes ten. Example 2: A decreasing sequence, a1 = 20, d = −3, n = 5. Not every sequence climbs. With a first term of 20 and a common difference of negative 3, the sequence is 20, 17, 14, 11, 8. The fifth term is a5 = 20 + (5 − 1) × (−3) = 20 − 12 = 8. The sum is S5 = 5/2 × (2 × 20 + (5 − 1) × (−3)) = 2.5 × (40 − 12) = 2.5 × 28 = 70, and adding the list gives 20 + 17 + 14 + 11 + 8 = 70. Decreasing sequences appear whenever something is used up steadily: a battery losing the same charge each hour, a debt reduced by the same payment each month, a countdown. The formulas do not care about the sign of d; they work identically, and the calculator shows the negative difference in the working so the subtraction is visible rather than hidden.

Example 3: Adding 1 to 100, a1 = 1, d = 1, n = 100. The most famous arithmetic series story concerns the young Carl Friedrich Gauss, who is said to have added the numbers from 1 to 100 in seconds by pairing them: 1 with 100 makes 101, 2 with 99 makes 101, and so on, giving fifty pairs of 101, or 5050. That pairing is exactly what the sum formula does. With a1 = 1, d = 1, n = 100, the hundredth term is a100 = 1 + 99 × 1 = 100, and the sum is S100 = 100/2 × (1 + 100) = 50 × 101 = 5050. The formula is Gauss's trick, generalised. It works because in an arithmetic sequence the first and last terms always pair to the same total as the second and second-last, and so on, so the average term is simply the average of the first and last, and the sum is the number of terms times that average. That average-term insight is the most useful way to remember the formula: sum equals how many terms, times the average term, and the average term of an arithmetic sequence is the midpoint of its first and last terms. Example 4: A savings plan, a1 = 5, d = 3, n = 20. Suppose you save 5 pounds in the first week and add 3 pounds more each week than the week before: 5, 8, 11, and so on. After twenty weeks, the weekly amount is a20 = 5 + 19 × 3 = 62 pounds, and the total saved is S20 = 20/2 × (2 × 5 + 19 × 3) = 10 × (10 + 57) = 670 pounds. This is the shape of many real plans that grow steadily rather than staying flat, and it shows why the sum matters as much as the last term. The final weekly amount, 62, sounds modest; the total, 670, is the number that decides whether the plan meets its goal. Any plan that increases by a fixed amount each period is an arithmetic sequence in disguise, and its total is an arithmetic series, whether or not anyone uses those words for it.

Why the Formulas Work

The nth term formula was explained above by counting steps, but the sum formula deserves its own short proof, because a formula you can prove is one you will remember. Write the sum forwards: S = a1 + (a1 + d) + (a1 + 2d) + … + an. Now write it backwards: S = an + (an − d) + (an − 2d) + … + a1. Add the two lines term by term. Each pair adds to a1 + an: the first pair is a1 + an, the second is (a1 + d) + (an − d) = a1 + an, and every pair is the same. There are n pairs, so 2S = n × (a1 + an), and dividing by two gives S = n/2 × (a1 + an). Substituting an = a1 + (n − 1) × d gives the form the calculator shows. No step in that argument depends on the values, which is why the formula is universal for arithmetic sequences, and why it fails for sequences that are not arithmetic: if the step is not constant, the pairs do not all add to the same total, and the trick collapses. That failure is itself a useful diagnostic. If you are given a list and you are not sure whether it is arithmetic, check the differences between consecutive terms. In 3, 7, 11, 15, the differences are 4, 4, 4: constant, so the sequence is arithmetic and the formulas apply. In 3, 6, 12, 24, the differences are 3, 6, 12: not constant, so it is not arithmetic at all; it is geometric, with each term doubling, and it needs different formulas. In 2, 4, 7, 11, the differences are 2, 3, 4: changing steadily, which signals a quadratic pattern, again a different tool. Checking differences first takes seconds and prevents the commonest misuse of this calculator, which is applying arithmetic formulas to a list that merely looks regular.

Sequences in Real Life

Arithmetic sequences turn up wherever change is steady. A taxi fare with a fixed charge per kilometre, a staircase with equal risers, a row of seats that gains the same number of chairs in each row, a loan repaid in equal instalments of principal, a temperature falling by the same number of degrees each hour: each is an arithmetic sequence, at least approximately. The word approximately matters, because real life rarely holds a step perfectly constant. A runner's weekly distance may increase by roughly the same amount, but weather and fatigue intervene. The calculator is exact for the ideal sequence you enter, and its value for real plans is as a model: put in the intended step, read off where it leads, and then judge whether the real world is likely to follow the model closely enough for the answer to guide a decision. There is also a deeper connection worth knowing. An arithmetic sequence is a linear function in disguise. If you plot term number on the horizontal axis and term value on the vertical axis, the points fall exactly on a straight line, with slope equal to the common difference and intercept related to the first term. That is why arithmetic sequences appear in the study of linear growth, and why their sums grow quadratically: adding a linear sequence gives a quadratic total, just as the sum 1 + 2 + … + n = n(n + 1)/2 is quadratic in n. Seeing that link helps when sequences stop being a standalone topic and start appearing inside algebra, functions, and calculus, where the same ideas return in more general forms.

Limits: What This Tool Does and Does Not Do

This tool works with arithmetic sequences only. It does not handle geometric sequences, where each term is multiplied by a fixed ratio, Fibonacci-style sequences, where each term depends on the two before it, or any sequence defined by a complicated rule. If your list's differences are not constant, the answers this tool gives for your first term and a guessed difference will be wrong, and the difference check described above is the way to catch that before you trust anything. The tool also caps n at 10000 and shows at most the first 50 terms in its list, for readability. The nth term and the sum are computed for the full n you enter, even when the list is truncated, so a large n still gives exact answers for the two quantities that matter most. Decimal first terms and differences are accepted, because real sequences are not always whole numbers, but n itself must be a whole number: there is no such thing as the 3.5th term of a sequence. If you enter a decimal n, the tool asks for a whole number rather than rounding silently, because silent rounding would answer a different question from the one you asked.

Privacy: Your Numbers Stay in Your Browser

Every number you enter into this arithmetic sequence calculator stays in your browser. Nothing is uploaded to a server, nothing is stored, and there is no account to create. The calculation runs entirely on your own device. That matters even for a maths tool, because the numbers people enter can be part of homework, coursework, a savings plan, or a piece of work they would rather keep private. Closing the page removes your inputs, and no one else sees them. This browser-only design is the same principle behind the other free tools on Your Smart Kit: a simple tool that does one job well, keeps your data private, and gives an answer you can check.

Frequently Asked Questions

What is an arithmetic sequence? An arithmetic sequence, or arithmetic progression, is a list of numbers in which each term after the first is found by adding a fixed number, the common difference, to the previous term. Examples are 3, 7, 11, 15 with difference 4, and 20, 17, 14, 11 with difference negative 3. What is the formula for the nth term? The nth term is an = a1 + (n − 1) × d, where a1 is the first term, d is the common difference, and n is the position of the term. The n − 1 counts the steps between the first term and the nth term. What is the formula for the sum of an arithmetic series? The sum of the first n terms is Sn = n/2 × (2a1 + (n − 1) × d), equivalently Sn = n/2 × (a1 + an). In words: the number of terms times the average of the first and last terms. What is the difference between a sequence and a series? A sequence is the list of terms; a series is the sum of those terms. The sequence 3, 7, 11, 15 has the series sum 36. This calculator reports both the terms and their sum. How do I know if a sequence is arithmetic? Subtract each term from the next. If all the differences are the same number, the sequence is arithmetic and this tool applies. If the differences change, it is not arithmetic, and you need a different method. Is the tool really free, and do I need an account? Yes, it is free, and no account is needed. There is no sign-up, no download, and no data uploaded. Open the page, enter the first term, the common difference, and how many terms to count, press "Calculate", and read the answers. You can use it as often as you like, for homework, revision, checking work, or planning anything that grows by steady steps.

Related Tools on Your Smart Kit

If you found the arithmetic sequence calculator useful, you may also like the other free calculators on Your Smart Kit, including the mean median mode calculator for summarising a set of numbers, the percentage calculator for working out shares and changes, and the step by step multiplication calculator for seeing every stage of a multiplication. Each runs in your browser, free, with no sign-up, and each is built on the same principle: a simple tool that does one job well, keeps your data private, and gives an answer you can check. Try the arithmetic sequence calculator at the top of this page with your own first term and difference, and see how far a single steady step can take a list.