Arithmetic Sequence Calculator Explained: From the Formula to the Worked Example

What an arithmetic sequence actually is

An arithmetic sequence is a list of numbers in which the gap between consecutive terms is the same fixed value. That gap is the common difference, almost always written as d. The starting value is the first term, written a₁. Once you know a₁ and d, every other term is determined: a₂ = a₁ + d, a₃ = a₁ + 2d, and so on. The Arithmetic Sequence Calculator takes a₁, d, and the number of terms n and returns the full list, the nth term aₙ, and the running total Sₙ in one pass.

The defining property is the constant difference. The list 3, 7, 11, 15, 19 is arithmetic because each step adds exactly 4. The list 3, 6, 12, 24 is not arithmetic because each step multiplies by 2 instead. Sequences with a constant ratio are geometric, not arithmetic, and they follow a different pair of formulas. The Geometric Sequence Calculator handles that case.

The two formulas you actually need

Every output the calculator returns reduces to two short formulas. They are both worth memorising because the same arithmetic powers the worked examples below, the FAQ answers, and almost every school-level sequence question you will see.

Nth term:  aₙ = a₁ + (n − 1) d
Sum of n terms (form 1):  Sₙ = n/2 × (a₁ + aₙ)
Sum of n terms (form 2):  Sₙ = n/2 × (2 a₁ + (n − 1) d)

The two sum forms are equivalent — the second is what you get when you substitute aₙ = a₁ + (n − 1)d into the first. Use form 1 when you already know the last term aₙ, and form 2 when you only know a₁, d, and n. The calculator works out both and cross-checks them internally, which is why a 1,000-term sequence resolves instantly: nothing is summed term by term, the closed-form Sₙ is evaluated directly.

These formulas live in chapter 9 of OpenStax's Algebra and Trigonometry and in essentially every secondary-school textbook in the world. The reason every textbook covers them is that they are the simplest non-trivial example of a closed-form replacement for a recursive definition. The recursive definition says "to find a term, keep adding d to the previous term." The closed form says "skip the recursion — here is the answer directly."

Where the formulas come from

The nth term is just counting d's

Start at a₁ and ask how many times you have added d by the time you reach the nth term. The first term has had d added zero times. The second term has had d added once. The third, twice. By induction, the nth term has had d added (n − 1) times. That is the entire derivation of aₙ = a₁ + (n − 1)d. The off-by-one — (n − 1) instead of n — is the most common mistake students make on this formula.

The sum formula and the Gauss schoolroom story

The sum formula is older than algebra and is usually credited to a young Carl Friedrich Gauss. The story, told in various forms, has a primary-school teacher setting the class the task of adding 1 + 2 + 3 + … + 100 to buy himself a quiet hour. A seven-year-old Gauss is supposed to have written down 5,050 within seconds.

The trick: pair the first and last term, then the second and second-to-last, and so on. The pairs all add to the same value. 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, … There are 50 such pairs across 100 numbers, so the total is 50 × 101 = 5,050. The general version is the closed-form sum: pair a₁ with aₙ, a₂ with aₙ₋₁, and so on. Every pair sums to (a₁ + aₙ), and there are n/2 pairs, giving Sₙ = n/2 × (a₁ + aₙ).

The pairing argument works whether n is even or odd. For odd n the middle term pairs with itself, but the algebra still produces n/2 × (a₁ + aₙ) — the fraction n/2 is doing the right work. A neat way to see it is to write the sum twice, once forwards and once backwards, then add column by column:

Sₙ  =  a₁     + a₂     + a₃     + … + aₙ₋₁  + aₙ
Sₙ  =  aₙ     + aₙ₋₁   + aₙ₋₂   + … + a₂    + a₁
────────────────────────────────────────────────
2Sₙ =  (a₁+aₙ) + (a₁+aₙ) + … + (a₁+aₙ)   (n times)
2Sₙ =  n × (a₁ + aₙ)
Sₙ  =  n/2 × (a₁ + aₙ)

That is the same derivation the OpenStax textbook gives, and is the cleanest one-line proof for a classroom.

Worked example 1: positive difference

Take a₁ = 2 and d = 3 with n = 10 terms. Run those through the Arithmetic Sequence Calculator and you get the list 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.

The tenth term is a₁₀ = 2 + (10 − 1) × 3 = 2 + 27 = 29. That matches the last item in the list, as it should. The sum is S₁₀ = 10/2 × (2 + 29) = 5 × 31 = 155. Cross-check with the second sum form: S₁₀ = 10/2 × (2 × 2 + 9 × 3) = 5 × 31 = 155. Both forms agree, and if you actually add the ten numbers by hand you also get 155.

Worked example 2: negative common difference

Sequences are allowed to decrease. Take a₁ = 100 and d = −7 with n = 8. The list is 100, 93, 86, 79, 72, 65, 58, 51. The eighth term is a₈ = 100 + (8 − 1) × (−7) = 100 − 49 = 51, which agrees with the last item in the list.

The sum is S₈ = 8/2 × (100 + 51) = 4 × 151 = 604. The same pairing logic that works for an increasing sequence works for a decreasing one because the pair sums are still constant: 100 + 51 = 151, 93 + 58 = 151, 86 + 65 = 151, 79 + 72 = 151 — four pairs of 151, total 604.

A common difference of zero is also legal and produces a constant sequence in which every term equals a₁. The sum collapses to Sₙ = n × a₁, which is exactly what the general formula gives once you set d = 0.

Worked example 3: fractional terms

Take a₁ = 0.5 and d = 0.25 with n = 12. The list is 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25. The twelfth term is a₁₂ = 0.5 + 11 × 0.25 = 0.5 + 2.75 = 3.25, and the sum is S₁₂ = 12/2 × (0.5 + 3.25) = 6 × 3.75 = 22.5. Decimals and negatives present no difficulty for either formula; the only restriction is that n itself must be a whole number.

Where arithmetic sequences appear in the real world

Simple interest

A savings account that pays a fixed cash amount per period rather than a percentage produces an arithmetic sequence of balances. Deposit £1,000 at simple interest of £50 per year and the balance after each year is 1,050, 1,100, 1,150, … — a₁ = 1,050 and d = 50. Compound interest behaves differently because the per-period addition itself grows; for that the Geometric Sequence Calculator is the right tool.

Straight-line depreciation

Accounting depreciates an asset by a fixed amount each year under the straight-line method. A £24,000 vehicle written down by £3,000 per year produces balances 24,000, 21,000, 18,000, … — a decreasing arithmetic sequence with d = −3,000. Tax authorities almost everywhere allow this method for at least some asset classes.

Seating, staircases, brickwork

Stadium seating where each row holds a fixed number more seats than the row below is the canonical example. A row with 30 seats on the front rank, growing by 4 seats per row over 20 rows, holds Σ from 30 + 4 × 0 to 30 + 4 × 19, a total of S₂₀ = 20/2 × (30 + 106) = 10 × 136 = 1,360 seats. The same arithmetic applies to staircases (each step is at a constant height above the last) and certain brickwork patterns.

Time tables and scheduling

Any train or bus running on a fixed-interval timetable generates an arithmetic sequence of departure times. The 14 trains per hour at 4-minute intervals leave at minute 0, 4, 8, 12, … with a₁ = 0 and d = 4. Finding the time of the nth train uses aₙ = a₁ + (n − 1)d directly.

Common mistakes

Treating (n − 1) as n

The biggest source of errors. Students see "the nth term" and instinctively write a₁ + n × d, which gives the (n + 1)th term, not the nth. Sense-check: at n = 1 the formula must return a₁. If you plug n = 1 into a₁ + n × d you get a₁ + d, which is wrong. Plug n = 1 into a₁ + (n − 1)d and you get a₁ + 0 × d = a₁, which is right.

Confusing the sequence with the series

The terms a₁, a₂, a₃, … are the sequence. The cumulative total a₁ + a₂ + … + aₙ is the series. The calculator returns both, but exam questions can ask for either, and students sometimes give the sum when the term was asked for or vice versa.

Forgetting that d can be negative or fractional

A sequence can decrease, stay constant, or step by fractional amounts. None of those break the formulas. If you are stuck because your common difference is −0.5 or 7/3, plug it in anyway. The arithmetic carries through cleanly.

Using the arithmetic sum on a geometric series

The pairing trick only works because the common difference is constant, so every pair (a_k + a_{n−k+1}) hits the same value. In a geometric sequence the terms grow multiplicatively and pair sums are not constant — the trick fails. For geometric series use Sₙ = a₁ × (1 − rⁿ) / (1 − r).

When the calculator is not enough

For one-off school problems, the calculator and the closed-form formulas are the whole job. For more elaborate questions you may need:

• The mean of an arithmetic sequence equals (a₁ + aₙ) / 2 — the average of the first and last terms. The Average Calculator handles arbitrary lists.
• Percentage change between terms shrinks as the sequence grows, because d is fixed but the base value increases. The Percentage Calculator handles that conversion.
• A mix of arithmetic and geometric structure — for example, a sequence that grows by a fixed cash amount plus a fixed percentage each period — needs either a spreadsheet or the closed-form sum of an arithmetico-geometric series, which is beyond the scope of either single-purpose calculator.

For more general sequence work where you are not yet sure whether the pattern is arithmetic, geometric, or something else, the Number Sequence Calculator accepts both kinds of input and reports the structure.

Frequently asked questions

What is the difference between an arithmetic sequence and an arithmetic series?

A sequence is the list of terms: 2, 5, 8, 11, 14. A series is what you get when you add them up: 2 + 5 + 8 + 11 + 14 = 40. The Arithmetic Sequence Calculator returns both — the listed terms and the partial sum Sₙ. In most textbooks "arithmetic progression" (AP) is used as a synonym for arithmetic sequence.

Why is the formula (n − 1)d and not nd?

Because the first term has no common difference added to it yet. a₁ = a₁ + 0 × d. By the time you reach a₂ you have added d once, a₃ twice, and aₙ a total of (n − 1) times. The off-by-one is the most common arithmetic-sequence mistake in school exams, and the sense-check at n = 1 is the quickest way to spot it.

How do I find the common difference if I only know two terms?

If you know aₘ and aₙ with m < n, then d = (aₙ − aₘ) / (n − m). For example, if a₃ = 11 and a₈ = 31, then d = (31 − 11) / (8 − 3) = 20 / 5 = 4. Plug d back into aₙ = a₁ + (n − 1)d to recover a₁ = 11 − (3 − 1) × 4 = 3, and the sequence is 3, 7, 11, 15, 19, 23, 27, 31, …

Can the partial-sum formula be used for an infinite arithmetic series?

No. An infinite arithmetic series diverges unless every term is zero (d = 0 and a₁ = 0). With any nonzero common difference the terms grow without bound in one direction or the other, so the running total grows without bound as well. Infinite sums are only finite for sequences whose terms shrink fast enough — like a geometric sequence with |r| < 1.

Is "1, 3, 5, 7, 9" an arithmetic sequence?

Yes — the common difference is 2. So are the even numbers (d = 2), the multiples of any integer (d = the integer itself), and every linear function evaluated at consecutive integers, since f(n) = mn + c produces terms with constant difference m. Arithmetic sequences are exactly the integer-input outputs of linear functions.

Does the calculator handle decimal or negative inputs?

Yes. a₁ and d can be any real number, including 0.5, −2.75, or 3.14. The number of terms n must be a whole number between 1 and 1,000 because partial fragments of a term are not meaningful and longer sequences become unreadable in a browser. For n above 1,000 the closed-form aₙ and Sₙ can still be computed by hand from the formulas above.

What is the famous Gauss story about adding 1 to 100?

A young Carl Friedrich Gauss is said to have been set the task of adding 1 + 2 + … + 100 by a schoolmaster who wanted a quiet classroom. Gauss noticed that pairing the first and last term gives 1 + 100 = 101, the second and second-to-last give 2 + 99 = 101, and so on — fifty pairs each summing to 101, total 5,050. The same pairing trick is the derivation of Sₙ = n/2 × (a₁ + aₙ).

Where do arithmetic sequences show up outside of math class?

Anywhere a quantity grows by a fixed amount each step: simple-interest balances, stadium seating where each row has a fixed number more seats than the row below, straight-line depreciation, fixed-interval timetables, and physical objects like the steps of a staircase or the rungs of a ladder. They are the discrete-time analogue of a linear function.