Add the same amount each step and you've built a sequence — a staircase of numbers. The magic? One short rule can teleport you to the 100th step without climbing the 99 below it.
Climb inA sequence is just a list of numbers in order. When you make that list by adding the same amount each step — like 5, 7, 9, 11 — you've built an arithmetic sequence: a perfectly even staircase.
Each number in the list is called a term. The first number is the first term, the next is the second term, and so on. The steady amount you add to get from one term to the next has a name too: the common difference. In 5, 7, 9, 11 the common difference is +2, because every step climbs by exactly 2. That's the whole personality of an arithmetic sequence — a starting point and a step that never changes.
Here's where it gets clever. If the staircase always rises by the same step, you shouldn't have to walk it one stair at a time to reach a far-off term. There ought to be a rule — a little formula — that lets you point at any position and instantly read off the term that lives there. That rule is called the nth term, and finding it is what this whole page is about.
There are two honest ways to describe an arithmetic sequence, and the difference between them is the most useful idea on this page.
The first is a term-to-term rule: it tells you how to get from one term to the next one. For 3, 6, 9, 12 the term-to-term rule is simply "add 3." It's friendly and true — but a little needy. To find term 100 with "add 3," you'd have to start at term 1 and add 3 ninety-nine separate times. Miss one and the whole answer is wrong.
The second is the position-to-term rule, better known as the nth term. The little n stands for a position in the list — n = 1 means the first term, n = 2 means the second, n = 100 means the hundredth. The nth term rule turns a position straight into its term, with no need to know the term before it. For 3, 6, 9, 12 that rule is 3n: feed it n = 100 and it hands you 3 × 100 = 300 in a single breath.
A term-to-term rule makes you climb. The nth-term rule lets you jump to any stair you like.
When a question says "find the nth term," it is always asking for the second kind — the teleporting rule. So that's the one we'll learn to build.
Every arithmetic sequence has an nth-term rule of the same shape: multiply the position by some number, then add a fixed amount. Here's the two-step way to find it, no heavy algebra needed.
Move 1 — the step is the multiplier. Find the common difference. In 5, 7, 9, 11 it's 2, so the rule begins with 2n. Why? Because each time n grows by 1, the term should grow by the step — and "2n" grows by exactly 2 every time. The step is the number in front of n. Always.
Move 2 — fix the start. Test your bare rule at n = 1. Plain 2n gives 2 × 1 = 2, but the sequence actually starts at 5. So 2n is 3 too small everywhere. Add that 3 back: the rule is 2n + 3. That "+ 3" is exactly the first term minus the common difference (5 − 2 = 3) — the little nudge that slides the whole staircase up to where it really begins.
Check it the way a careful person would. Position 1: 2 × 1 + 3 = 5. ✓ Position 4: 2 × 4 + 3 = 11. ✓ It matches every term you can see, so you can trust it for the ones you can't — like position 100: 2 × 100 + 3 = 203, found in one line.
Quick warning we'll return to: the step gives you the multiplier, but almost never the whole rule. You usually still need that "+ something" fix-up. Forgetting it is the single most common slip — the next demo lets you feel both pieces move.
This is the heart of the page. Pick a starting sequence, or set the first term and the step (common difference) with the sliders. Watch the staircase grow, watch the rule build itself underneath — and then type any position into Jump straight to term and get that term instantly, no counting.
The teal staircase shows the first six terms; each riser is the same height because the step never changes. The rule below is built live: the step becomes the number in front of n, then a fix-up lines up the start. The glowing stair is the term you jumped to.
Play for a minute and a feeling sets in. The shape of the rule never changes — it's always "step × n, then a fix-up." Only the two dials change. Slide the difference and the staircase tilts steeper or gentler; slide the first term and the whole staircase rides up or down. The rule reads your dials and rewrites itself every time.
Let's do the two moves slowly, by hand, on the two sequences from the demo. Follow along and you'll have the method for good.
Move 1: the step from one term to the next is +3, so the rule begins with 3n. Move 2: test 3n at position 1 — it gives 3 × 1 = 3, which is already the first term. The fix-up is 0 (because first term − step = 3 − 3 = 0), so nothing to add. The rule is simply 3n. Check: position 4 → 3 × 4 = 12. ✓ Want term 100? 3 × 100 = 300. Done.
Move 1: the step is +2, so the rule begins with 2n. Move 2: test 2n at position 1 — it gives 2, but the sequence starts at 5, so it's 3 too small. The fix-up is first term − step = 5 − 2 = 3. Add it: the rule is 2n + 3. Check: position 1 → 2 + 3 = 5 ✓, position 4 → 8 + 3 = 11 ✓. Term 100? 2 × 100 + 3 = 203.
Notice the pattern across both. The step decided the multiplier (3 then 2). The start decided the fix-up (0 then +3). Two dials, every time. And it works downhill too: for 20, 17, 14, … the step is −3, so the rule contains −3n, and the fix-up is 20 − (−3) = 23, giving −3n + 23. A staircase going down is just an up-staircase with a negative step; the recipe never changes.
Imagine someone asks for the 100th term of 5, 7, 9, 11, … With only "add 2," you're stuck climbing: 5, 7, 9, 11, 13 … one stair at a time, ninety-nine additions, plenty of chances to lose count. With the rule 2n + 3, you write one line — 2 × 100 + 3 = 203 — and you're there. Same answer, a fraction of the work, and far harder to get wrong.
That's the real prize of the nth term. It cuts the rope that ties each term to the one before it. Every term suddenly stands on its own position, ready the instant you ask for it — term 7 or term 7,000, the rule doesn't care how far away you point. You've turned a long walk into a teleport.
A rule is a machine for skipping the boring middle.
Flip it around. Here's a mystery arithmetic sequence — you hunt for its nth term. Set the two dials: how much to multiply n by, and what to add after. Each cell turns teal when your rule matches and red when it doesn't. Get all five teal and you've caught it.
Tip from the recipe: read the step between terms and set Multiply n by to that first. Then nudge Then add until position 1 lines up. Once it's all teal, the rule even predicts term 50 — far past anything shown.
These three mix-ups catch almost everyone at first. Spot them once and you'll dodge them forever.
The step does give you the multiplier — but rarely the whole rule. For 5, 7, 9, 11 the step is 2, yet plain 2n gives 2, 4, 6, 8 — every term is 3 too small. You still need the "+ b" fix-up (here +3, so 2n + 3). The rule is only "just the difference" when the first term happens to equal the difference, like 3, 6, 9 → 3n. Always test position 1 and patch the gap.
The little n is a position — 1st, 2nd, 3rd — not a term value. You feed the rule a position and it hands back the term that lives there. n = 100 doesn't mean "the term 100"; it means "the 100th term," which here is 2 × 100 + 3 = 203. Keep "position in, term out" straight and the rest is easy.
Nope — that's exactly the climbing you built the rule to skip. "Add 2" forces you up every stair; the nth term jumps straight to any position in one calculation. If you ever catch yourself listing terms to reach a far one, stop and build the rule instead. One multiply and an add beats a hundred steps.
Last challenge. For each sequence, pick its nth-term rule. Use the recipe — find the step, then check position 1. There's an explanation after every answer, so a wrong guess is just a faster way to learn.
Every sequence here is arithmetic: step × n, then a fix-up. If the step is 3, the rule contains 3n — but always double-check what to add or subtract.
Arithmetic sequences aren't only a school exercise — anything that grows by the same amount each time is one, with its own nth term quietly running underneath.
Row 1 has 12 seats and each row adds 2. Row n holds 2n + 10, so row 30 has 70 — no counting rows.
Start with $5, add $3 a week. After n weeks you've got 3n + 5 dollars. Week 20? That's $65.
Each step rises the same amount. The top of stair n is a fixed start plus n equal risers — pure arithmetic.
In every case the win is identical: instead of listing forever, you capture the whole staircase in one short rule. That little n standing for "any position" is your first real taste of algebra — and once you can write rules like 2n + 3, you're ready to start solving them: "which seat row holds exactly 50 people?" becomes a puzzle you can answer.
The common difference — the steady amount added each time — becomes the number in front of n.
Test at position 1 and add (first term − step) so the rule lands on the real first term.
Feed the rule any position and it hands you that term instantly — no climbing required.