Arithmetic Sequence Calculator — nth term and series sum

The arithmetic sequence calculator above finds any term of a sequence that grows by a fixed amount each step, adds up the first n terms, and works backwards to tell you where a particular value sits in the sequence. It prints the substituted formula and the opening terms so you can check the pattern by eye.

Arb Digital publishes this as part of a free maths toolkit. Arithmetic sequences are the simplest kind of pattern and the most widely used: seat rows in a theatre, fixed repayment schedules, depreciation by a flat amount, stacked pipe counts and simple interest all follow the same two-parameter structure.

What This Arithmetic Sequence Calculator Does

Enter the first term and the common difference — the fixed amount added at each step — and the tool returns the nth term as the headline figure. Alongside it you get the sum of the first n terms, the mean of those terms, the total span from the first to the nth, and the position of any target value you want to locate in the sequence.

The position search is the genuinely useful extra. Rather than counting forward until you hit a value, the calculator solves for n directly and tells you whether the value is actually in the sequence at all. A value that lands between terms returns a fractional position, which the tool reports as "not a term" along with the two positions it sits between — the answer most tools simply refuse to give.

A preview of the opening terms is printed under the working line, so a mistyped difference is obvious immediately rather than after the arithmetic.

How to Use It

1. Enter the first term. This is a₁, the value at position 1, not position 0. Off-by-one errors here shift every subsequent result.
2. Enter the common difference. Positive for a rising sequence, negative for a falling one, and zero for a constant sequence, which is still valid.
3. Choose the term number n. The nth term and the sum of the first n terms are both computed from this.
4. Add a target value if you want a position. The tool solves for n and tells you if the value is not actually a term.
5. Check the sequence preview under the results to confirm the pattern is the one you intended.

The Formula / How It's Calculated

An arithmetic sequence is defined by two numbers: a starting value a₁ and a common difference d. Every term is the previous one plus d, which makes the nth term a straight-line function of n:

aₙ = a₁ + (n − 1)d

The bracket is where marks are lost. Reaching the 20th term takes 19 steps, not 20, because the first term is already there before any step is taken. With a₁ = 5 and d = 3, the 20th term is 5 + 19 × 3 = 5 + 57 = 62.

The sum of the first n terms — the arithmetic series — has an elegant form:

Sₙ = (n ÷ 2) × (a₁ + aₙ), equivalently Sₙ = (n ÷ 2) × (2a₁ + (n − 1)d)

For the same sequence: S₂₀ = (20 ÷ 2) × (5 + 62) = 10 × 67 = 670. The reason this works is the pairing trick: write the series forwards and backwards under itself, and every vertical pair adds to the same total, a₁ + aₙ. There are n such pairs, counted twice, hence the division by two. MathWorld's entry on the arithmetic series sets out the derivation formally.

Why the Sum Is n Times the Average Term

Look at Sₙ = (n ÷ 2) × (a₁ + aₙ) again and rearrange it: it is n multiplied by the average of the first and last terms. For an arithmetic sequence — and only for one — the mean of the first and last terms equals the mean of all the terms, because the values are evenly spaced around their own centre.

That gives you a fast mental method. To add the numbers 1 to 100, take the average of 1 and 100, which is 50.5, and multiply by 100 terms: 5,050. The story that a young Carl Friedrich Gauss startled his teacher with this shortcut is well known in mathematics teaching, and whatever its historical accuracy, the technique itself is exactly the pairing argument above.

The same reasoning is why the mean of an evenly spaced dataset equals its median. It fails the moment the spacing is uneven — for a geometric sequence, where each term multiplies rather than adds, the correct centre is the geometric mean and the sum formula is completely different.

Finding the Difference From Two Known Terms

You often know two terms but not the difference. Subtract the values, divide by the gap in positions:

d = (aₚ − a_q) ÷ (p − q)

If the 4th term is 14 and the 9th is 29, then d = (29 − 14) ÷ (9 − 4) = 15 ÷ 5 = 3. From there, work back to the first term: a₁ = a₄ − 3d = 14 − 9 = 5. The whole sequence is then determined, and you can enter those two values in the calculator to check the rest.

This is the standard exam question type, and it is also how you test whether real data is arithmetic at all. Take differences between consecutive values: if they are constant, the sequence is arithmetic. If the ratios are constant instead, it is geometric and grows by percentage rather than by amount — a distinction the percentage change calculator makes easy to check.

Arithmetic Versus Geometric: The Difference That Matters

An arithmetic sequence adds a fixed amount each step, so it plots as a straight line. A geometric sequence multiplies by a fixed factor, so it curves upward or decays. Over a few terms they can look similar; over many, they are not comparable at all.

Start at 100. Adding 10 each year gives 200 after ten years. Multiplying by 1.10 each year gives 259.37 — and after fifty years the arithmetic version reaches 600 while the geometric one passes 11,700. The sum of a geometric sequence follows an entirely different rule, set out in MathWorld's entry on the geometric series, and no amount of adjustment turns one formula into the other. This is exactly the difference between simple and compound growth, and mistaking one for the other is the single most consequential error in any projection. If your steps are a fixed percentage, the sequence is geometric and needs the powers handled by the exponent calculator, not the linear formula on this page.

Negative and Zero Common Differences

A negative difference gives a decreasing sequence, and the formulas need no modification whatsoever: with a₁ = 100 and d = −7, the 10th term is 100 + 9 × (−7) = 37. A sequence like this eventually goes negative, which may or may not be meaningful depending on what it represents — stock levels and depreciation schedules usually stop at zero in reality, even though the mathematics happily continues.

A difference of zero gives a constant sequence where every term is a₁. This is technically a valid arithmetic sequence, and the sum is simply n × a₁. It matters because the position search breaks down here: if d is zero, either every term equals the target or none does, so no single position exists. The calculator states this rather than dividing by zero, which is exactly the guard most implementations forget.

Where Arithmetic Sequences Show Up

Anywhere a fixed amount is added on a regular schedule. Theatre and stadium seating usually adds a constant number of seats per row, so the total capacity is an arithmetic series. Straight-line depreciation removes the same amount of value each year. A fixed-amount savings plan without interest accumulates arithmetically. Simple interest — as opposed to compound — adds the same amount each period by definition, which makes the balance an arithmetic sequence.

Stacked objects give a neat case: pipes or barrels arranged with one fewer in each ascending layer form a sequence with d = −1, and the total in the stack is the arithmetic series. The same counting logic underlies the triangular numbers 1, 3, 6, 10, which are the running sums of the sequence 1, 2, 3, 4 and appear throughout combinatorics. For factorial-based counting problems, which grow far faster than any arithmetic sequence, our factorial calculator is the right tool.

Common Mistakes to Avoid

• Using n instead of (n − 1) in the nth-term formula — the first term needs no steps, so the 20th term takes 19 of them.
• Counting terms inclusively the wrong way — the number of terms from position p to position q is q − p + 1, not q − p.
• Confusing the sequence with the series — the sequence is the list of terms, the series is their running total.
• Applying arithmetic formulas to percentage growth — a fixed percentage each period is geometric, and the linear formula will understate it badly.
• Assuming a target value must be in the sequence — solving for n can give a fraction, which means the value falls between two terms.

Related Free Tools From Arb Digital

Sequences that multiply rather than add need the powers in the exponent calculator, and the logarithm calculator solves for the step count when growth is geometric. Growth rates in percentage terms are handled by the percentage change calculator, counting problems by the factorial calculator, and general arithmetic by the scientific calculator. Ratio-based patterns are covered by the ratio calculator. Browse the full free online tools hub for more.

Frequently Asked Questions

This tool is provided for general educational use. Sequences that grow by a fixed percentage rather than a fixed amount are geometric and require different formulas.