Arithmetic Sequence Calculator

Why the exponent is n − 1

This is the single most common mistake, and it is worth being explicit about.

Term 1 is the first term. No addition has happened. Term 2 has had one addition, term 3 has had two. By term 20 there have been nineteen. The number of steps is always one fewer than the term number, because counting the terms starts at 1 while counting the steps starts at 0.

Use n instead of n − 1 and every answer is one common difference too large — 103 instead of 98. It is a consistent, invisible error: the answer looks plausible, which is exactly what makes it dangerous.

The sum, and Gauss’s trick

The story goes that Gauss, asked as a schoolboy to add the numbers from 1 to 100, produced 5050 almost at once. Whether or not it happened, the method is the one worth knowing.

Pair the first with the last: 1 + 100 = 101. The second with the second-last: 2 + 99 = 101. Every pair gives 101, and there are fifty of them. So the total is 50 × 101 = 5050.

That generalises to any arithmetic sequence, because the pairing always works: as one end goes up by d, the other comes down by d, so every pair has the same total. Hence S_n = n(a₁ + a_n)/2 — the average of the first and last term, multiplied by how many terms there are.

A pleasing consequence: the first n odd numbers sum to exactly n². Try 1, 3, 5, … with 100 terms and the total is 10,000.

Closed form or recursive

The same sequence can be described two ways, and both are on the page because different courses teach different ones.

The recursive form says how to get from one term to the next: a₁ = 3, a_n = a_n−1 + 5. Faithful to how the sequence is built, and useless for finding term 500 without finding the 499 before it.

The closed form jumps straight there: a_n= 3 + (n − 1)×5. Term 500 is one line of arithmetic.

For arithmetic sequences the closed form is easy to derive, which is why it is the one used. Not every sequence has one that simple — the Fibonacci numbers famously need the golden ratio to write theirs down.

Finding the rule from two terms

A common exam shape: you are told term 4 is 17 and term 9 is 42, and asked for the rule.

Five steps separate them, and they differ by 25, so the common difference is 25 ÷ 5 = 5. Then work backwards from term 4: three steps before it, so a₁= 17 − 3×5 = 2.

The general version: d = (a_m − a_n) / (m − n), and then substitute either known term to recover a₁. Once you have both, this page gives you every other term and the sum.

Where they turn up

Seating and stacking. A theatre with two extra seats per row, a stack of logs with one fewer on each layer — the total is an arithmetic sum, which is why the pairing formula appears in trade estimating as often as in classrooms.

Simple interest. Interest on the original amount only grows by a fixed sum each period, so a simple-interest balance is arithmetic. Compound interest is geometric, which is the whole reason it outruns it.

Depreciation on a straight line. Take a fixed amount off each year and the book value is an arithmetic sequence, which is what makes the straight-line method easy to audit.

Anything paced evenly. A training plan adding five minutes a week, a payment plan of equal instalments — if the step is constant, this is the sequence.

Why exact arithmetic matters here

Enter a first term of 0.1 and a common difference of 0.1, and ask for term 40. The answer is 4.

Add 0.1 forty times in floating point and you get 3.9999999999999996. Neither 0.1 nor the running total can be stored exactly in binary, and the error compounds with every addition.

This page carries the terms as exact fractions on arbitrary-precision integers, so 0.1 is really one tenth and term 40 is really 4. Fractions entered as 1/3 or 1/6 stay exact too, rather than becoming 0.3333333333.

For a homework answer the difference is cosmetic. For anything that gets summed a few thousand times, it is not — and the closed form exists precisely so the drift never happens.

Sources and methodology

Sequences are standard curriculum material; these are the references for how they are defined and assessed.

• National Council of Teachers of Mathematics — where sequence notation and the n − 1 convention are set out
• NIST SP 811 — the conventions for reporting exact and rounded numeric results
• Geometric Sequence Calculator — the multiplying counterpart
• How we calculate — methodology

Method. Terms and sums are carried as exact rationals on arbitrary-precision integers, so a common difference entered as 0.1 or as 1/6 stays exact however many terms are taken. The partial sum comes from the pairing formula rather than from a loop, and the suite checks it against a term-by-term loop across two thousand generated sequences — including the case where floating point would visibly drift. That engine is verified on every change against 76 hand-written assertions, including that a sequence starting at 0.1 with a step of 0.1 gives exactly 4 at term 40, which through doubles gives 3.9999999999999996. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

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Read the guide

If the terms multiply instead of adding, the Geometric Sequence Calculator is the one — and if you have the terms but not the rule, the Nth Term Calculator works backwards from them.