What makes a sequence arithmetic?

The gap between consecutive terms is the same all the way through. Subtract each term from the one after it: if you always get the same number, that number is the common difference and the sequence is arithmetic. If the gaps change, it is not — though if the ratios are constant instead, it is geometric. Checking three or four consecutive gaps is usually enough to tell.

Why is it (n − 1)d and not nd?

Because the first term has not taken a step yet. Getting to the second term takes one step, to the third takes two, and to the nth takes n − 1. Count the gaps in a written-out sequence rather than the terms and it becomes obvious: ten terms have nine gaps between them. Using nd gives a₁ + nd, which is the term after the one you wanted — the error is always exactly one position, in the same direction, every time.

How do I find the common difference from two terms?

Divide the change in value by the change in position: d = (aₘ − aₙ) ÷ (m − n). The terms do not have to be next to each other. If a₇ is 38 and a₃ is 18, the value rose by 20 across four positions, so d is 5. Then work back to the first term with a₁ = aₙ − (n − 1)d, which here gives 18 − 2 × 5 = 8. The only case that fails is two terms at the same position, where the division is by zero.

Why does the sum formula have n/2 in it?

Because it is counting pairs. Pair the first term with the last, the second with the second-last, and so on: every pair adds to the same total, since a step forward at one end cancels a step back at the other. With n terms there are n/2 pairs, each worth a₁ + aₙ. An odd number of terms leaves the middle one unpaired, but it happens to be exactly half of a pair’s value, so the formula still works.

Can the sum of an arithmetic sequence go down?

Yes, whenever the terms turn negative. The running total is not linear in n — it works out to (d/2)n² + (a₁ − d/2)n, a parabola. With a negative common difference the parabola opens downwards, so the total rises, reaches a maximum, and then falls as each new negative term is added. The terms themselves stay perfectly regular throughout; it is only their accumulation that turns.

Where is the sum largest?

At the last term that is still non-negative. Up to that point every term added something; after it every term takes something away. So the total peaks precisely as the sequence crosses zero, which you can find without calculus by asking when aₙ stops being positive. The one subtlety is a term landing exactly on zero: adding it changes nothing, so two consecutive positions give the same total and neither is more correct.

How do I tell whether a number is in the sequence?

Rearrange for the position: n = (value − a₁) ÷ d + 1. That number has to be a positive whole number for the value to be a genuine term. With a₁ = 3 and d = 5, the value 103 gives n = 21 and is a term; the value 20 gives n = 4.4 and is not — it sits between the fourth term, 18, and the fifth, 23, without ever being reached. Many calculators print the fractional index without saying it disqualifies the value.

What is the difference between arithmetic and geometric?

Arithmetic adds a fixed amount each step; geometric multiplies by a fixed amount. So 3, 8, 13, 18 is arithmetic with a difference of 5, while 3, 6, 12, 24 is geometric with a ratio of 2. Test by both: take differences and take ratios, and see which comes out constant. A sequence can be both only if it is constant — where the difference is zero and the ratio is one.

How many terms are there between two values?

Divide the total change by the step and add one: n = (last − first) ÷ d + 1. The plus one is the fencepost that gets forgotten — counting from 10 to 20 in steps of 2 gives 10, 12, 14, 16, 18, 20, which is six terms and not five, because both ends are included. The same reasoning explains why the nth term uses n − 1: there are always one fewer gaps than terms.

What happens if the common difference is zero?

Every term equals the first, and the sequence is still arithmetic — a constant gap of zero is a constant gap. The nth term formula collapses to aₙ = a₁, and the sum becomes simply n × a₁. It is also the only case where a sequence is both arithmetic and geometric, since multiplying by one and adding zero do the same thing. Asking whether a value appears has only two answers: it matches the constant or it does not.

Arithmetic Sequence Calculator

An arithmetic sequence adds the same amount each step. The nth term is a₁ + (n − 1)d, and the first n terms add to n/2 times (a₁ + aₙ). The (n − 1) is where nearly every mistake happens. The first term has taken no steps yet, so the fifth term is the first plus four differences, not five. Writing a...

It is (n − 1)d, not nd. The first term has taken no steps yet, so a₅ is a₁ plus four differences rather than five. Writing a₁ + nd returns the next term every time — it is the single commonest error in the topic, and it is always wrong by exactly one step.

THE SEQUENCE

a₁d

nhow many terms

aₙ = 3 + (n − 1)(5)

a10

S10

255

a10 takes 9 steps of 5 from 3, not 10. The sum is 10/2 × (3 + 48).

The terms. 3, 8, 13, 18, 23, 28, 33, 38, 43, 48

Quantity	Working	Result
Steps taken	n − 1 = 10 − 1	9
a10	3 + 9 × 5	48
S10, paired	10/2 × (3 + 48)	255
S10, expanded	10/2 × (2×3 + 9×5)	255
Mean term	S10 ÷ 10	25.5

THE SHAPE OF IT

The terms lie on a straight line and their running total does not. Each step adds the same amount, so the terms are linear in n; but the sum accumulates a growing quantity, which makes it quadratic. That is the whole reason a total can peak and then fall while the terms are still perfectly regular.

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HOW TO USE

1
Give the first term and the common difference, or give any two terms and let the page work them out. Two terms need not be adjacent — the difference is the change in value divided by the change in position.
2
Set how many terms you want. Positions are whole numbers, so a fractional n is rounded down rather than interpolated: there is nothing between the fourth term and the fifth.
3
Look at the running total, not just the last term. Where the sequence changes sign the total turns, and the page names the position where it does.
4
Use the third mode to test whether a particular value appears. The formula always returns a number; only a positive whole one means the value is genuinely in the sequence.

REFERENCE FORMULAS

Rule	Formula	What it is for
nth term	aₙ = a₁ + (n − 1)d	The (n − 1) is the whole difficulty. a₁ + nd gives the next term, not this one.
Common difference	d = aₙ₊₁ − aₙ	Constant by definition. If consecutive gaps differ, it is not an arithmetic sequence.
From two terms	d = (aₘ − aₙ) ÷ (m − n)	Works for any two terms, not just neighbours. Then back out a₁.
Sum, from both ends	Sₙ = n/2 (a₁ + aₙ)	Gauss’s pairing: first with last, second with second-last, each pair equal.
Sum, from a₁ and d	Sₙ = n/2 (2a₁ + (n − 1)d)	The same formula with aₙ substituted. Use when the last term is unknown.
Sum as a quadratic	Sₙ = (d/2)n² + (a₁ − d/2)n	A parabola in n — which is why the running total can have a turning point.
Index of a value	n = (aₙ − a₁) ÷ d + 1	Must come out a positive whole number, or the value is not in the sequence.
Sum of 1 to n	n(n + 1)/2	The case a₁ = 1, d = 1. Gauss reportedly found it aged nine.
Sum of first n odds	n²	a₁ = 1, d = 2. The odd numbers sum to perfect squares, exactly.
Arithmetic mean	(aₘ + aₙ)/2 = a₍₍ₘ₊ₙ₎/₂₎	The mean of two terms is the term halfway between, when that index exists.
Turning point	last n with aₙ ≥ 0, when d < 0	Adding negative terms shrinks the total, so the sum peaks as the terms cross zero.
Number of terms	n = (last − first) ÷ d + 1	Counting a range inclusively. The +1 is the fencepost that gets forgotten.
Not geometric	arithmetic adds d; geometric multiplies by r	Check differences for one, ratios for the other. Only a constant sequence is both.
Zero difference	d = 0 ⟹ aₙ = a₁ for all n	Still an arithmetic sequence. The sum is simply n × a₁.

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Last updated: August 4, 2026 · aₙ = a₁ + (n − 1)d, and Sₙ is quadratic in n · The running total can peak before the sequence does.