The Arithmetic Sequence Formula, Worked Out: nth Term, Sum, and Common Difference

An arithmetic sequence is the simplest kind of pattern in number theory: start somewhere, then add the same amount over and over. 2, 5, 8, 11 adds 3 each time. 20, 17, 14, 11 adds -3. That fixed step is the only thing you really need to know about a sequence, because once you have it, every other question — the 50th term, the sum of the first 100 terms, where a missing value sits — falls out of two short formulas. This guide walks through both, with numbers you can check by hand.

The Two Formulas You Actually Need

There are only two. The first gives you any term without writing out the ones before it:

aₙ = a₁ + (n − 1)·d

Here a₁ is the first term, d is the common difference (the fixed step), and n is the position you want. The part people trip over is the n − 1, not n. The first term has had the difference added zero times, the second term once, the third twice. So the multiplier always lags the position by one. For 2, 5, 8, 11 with a₁ = 2 and d = 3, the 6th term is 2 + (6 − 1)·3 = 2 + 15 = 17 — not 2 + 6·3 = 20.

The second formula adds up the first n terms in one shot:

Sₙ = n(a₁ + aₙ) / 2

Read it as: take the first term and the last term, average them, and multiply by how many terms there are. It works because pairing the first with the last, the second with the second-to-last, and so on, gives the same total every time. This is the trick attributed to a young Gauss, who supposedly summed 1 through 100 in seconds: 100·(1 + 100)/2 = 5050. If you don't yet know the last term, expand aₙ to get the equivalent form Sₙ = n(2a₁ + (n − 1)d) / 2, which only needs a₁, d, and n.

A Full Worked Example

Let me run one all the way through. Suppose the first two terms are 2 and 5, so the common difference is d = 5 − 2 = 3 and a₁ = 2.

Finding the 10th term. Plug into aₙ = a₁ + (n − 1)·d:

a₁₀ = 2 + (10 − 1)·3 = 2 + 9·3 = 2 + 27 = 29.

So the 10th term is 29. You can spot-check it by listing: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 — count them, the tenth is 29.

Summing the first 10 terms. Now that we know a₁ = 2 and a₁₀ = 29, use Sₙ = n(a₁ + aₙ)/2:

S₁₀ = 10·(2 + 29) / 2 = 10·31 / 2 = 310 / 2 = 155.

Add the listed run by hand if you want the proof: 2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 = 155. The pairing logic shows why: 2 + 29 = 31, 5 + 26 = 31, 8 + 23 = 31 — five pairs of 31 is 5·31 = 155.

That is the whole method. Difference first, then nth term, then sum.

Reading and Reverse-Solving the Common Difference

The common difference is the spine of the sequence, so getting it right matters more than anything else. To read it off a list, subtract any term from the one after it: in 2, 5, 8, 11 every step is +3. Crucially, check more than one gap. If 2, 5, 8, 12 gives gaps of 3, 3, 4, the steps are not equal and no single d describes the list — it isn't arithmetic at all, and any formula you apply will produce a wrong answer with confidence.

Sometimes a problem hands you two scattered terms instead of a clean list. Say you only know the 1st term is 2 and the 4th term is 11. The difference comes from:

d = (term₂ − term₁) / (number of steps between them)

The 1st and 4th terms are 3 steps apart, not 4 — you count gaps, not positions. So d = (11 − 2) / 3 = 9 / 3 = 3, which rebuilds 2, 5, 8, 11. Miscounting that gap as 4 is the single most common reverse-solving error; it scales d wrong and the whole reconstruction drifts. The arithmetic sequence calculator has a built-in reverse solver that takes the two terms and the step count and returns d for you, then feeds it straight into the nth-term and sum panels.

Negative, Fractional, and Zero Differences

The formulas don't care about sign or whether d is a whole number. A few cases worth knowing:

• Negative difference. a₁ = 20, d = -3 gives 20, 17, 14, 11 — a decreasing run. Watch the sum here: as terms turn negative the partial sum can fall instead of grow, which surprises people expecting every Sₙ to climb.
• Fractional difference. a₁ = 1, d = 0.5 gives 1, 1.5, 2, 2.5. Decimals are fine; just don't clear the field — type the 0.5.
• Zero difference. d = 0 produces a constant sequence where every term equals a₁, so the sum is simply n·a₁. It's a degenerate case, but the formula still holds.

The one mistake that quietly ruins answers in all three cases is confusing the common difference with a common ratio. Arithmetic sequences add a fixed d; geometric ones multiply by a fixed r. 2, 5, 8 keeps adding 3; 2, 4, 8 keeps multiplying by 2. If your pattern compounds rather than steps evenly, you want a geometric sequence calculator instead — plugging a geometric run into the arithmetic formula gives nonsense. And if you're staring at a list and aren't even sure which type it is, a number sequence generator lets you build out candidate patterns and compare the gaps directly.

Where This Shows Up

Arithmetic sequences hide inside a lot of everyday counting. A savings plan where you bank 100 the first month and add 25 more each month is a₁ = 100, d = 25; the 24th deposit is 100 + 23·25 = 675, and the two-year total is 24·(100 + 675)/2 = 9300. Fence posts spaced every 1.5 metres, salary steps that rise by a fixed raise, theatre seating that adds a fixed number of chairs per row, even the load on a staircase — all arithmetic. The same two formulas that grade a homework worksheet also size a layout or a budget.

The honest reason to use the formulas rather than a spreadsheet column is speed and trust: one line replaces a hundred rows of addition, and there's no place for an off-by-one to hide. Get the difference right, count steps not positions, remember the n − 1, and the rest is arithmetic in the most literal sense.

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Made by Toolora · Updated 2026-06-13