nth Term of an AP: Formula, Examples & How to Find It (Class 10)

In one line: the nth term of an AP is given by the formula aₙ = a + (n − 1)d, where a is the first term, d is the common difference and n is the position of the term you want.

So if you want the 10th term of 2, 7, 12, …, you don’t have to write out all ten numbers. Here a = 2 and d = 5, so a₁₀ = 2 + (10 − 1)×5 = 2 + 45 = 47. The formula lets you jump straight to any term — the 10th, the 50th, even the 500th — using only the first term and the common difference.

The nth term (also called the general term) is simply the term sitting in the nth position of the list. The idea behind the formula is easy: to reach the nth term you start at a and add the step d again and again. To get to the 2nd term you add d once, to the 3rd you add it twice, so to reach the nth term you add d exactly (n − 1) times — that is why it is (n − 1) and not n. The rest of this guide shows you how to use aₙ = a + (n − 1)d to find a term, work out which term a value is, rebuild an AP from two given terms, and test whether a number belongs to an AP.

🎬 Watch (60-sec Short): Can you find the 100th term of this AP? ⚡

This guide builds directly on the basics of what an AP is. If “first term” and “common difference” feel new, read What Is an Arithmetic Progression? first, then come back here.

The nth term of an AP formula

For an AP with first term a and common difference d, the nth term is:

aₙ = a + (n − 1)d

What it means in plain words: start at the first term, then add the common difference (n − 1) times to reach the term in position n.

What each symbol means:

• aₙ — the nth term, i.e. the term you are trying to find (the “answer”).
• a — the first term of the AP.
• d — the common difference (subtract any term from the term right after it: d = a₍ₖ₊₁₎ − aₖ).
• n — the position (or term number). It must be a positive whole number (1, 2, 3, …).

When it applies: for every arithmetic progression, finite or infinite. The only requirement is that the list really is an AP (equal common difference throughout).

Worked example. Find the 10th term of the AP 2, 7, 12, …

• First term: a = 2. Common difference: d = 7 − 2 = 5. Position: n = 10.
• aₙ = a + (n − 1)d → a₁₀ = 2 + (10 − 1)×5 = 2 + 45 = 47.

The 10th term is 47. Notice we multiplied d by 9, not 10 — because reaching the 10th term takes only 9 steps from the first. Getting this “(n − 1)” right is the single most important habit in this whole topic, and examfront’s Mistake Identification flags it the moment you slip into multiplying by n.

The last term of a finite AP is often written as l, and it is just a special nth term: l = a + (n − 1)d, where n is the total number of terms.

How to find a specific term of an AP

This is the most direct use of the formula: you are given the AP (or a and d) and a position, and you want that term.

Method: identify a, find d, put your value of n into aₙ = a + (n − 1)d, and simplify.

Worked example. Find the 20th term of the AP 2, 7, 12, …

• a = 2, d = 5, n = 20.
• a₂₀ = 2 + (20 − 1)×5 = 2 + 95 = 97.

That is the whole method — no listing, no counting on your fingers. Once the formula feels automatic, you can rattle off the 30th or 100th term just as quickly. Drilling a graded set of “find the term” questions on Topic Practice in examfront is the fastest way to make this second nature.

How to find which term of an AP equals a given value

Here the question is turned around: you know the value of a term and you want its position n. Because n is the only unknown, you solve a simple equation.

Method: set aₙ equal to the given value in aₙ = a + (n − 1)d, then solve for n.

Worked example. Which term of the AP 21, 18, 15, … is −81?

• a = 21, d = 18 − 21 = −3, aₙ = −81.
• −81 = 21 + (n − 1)(−3)
• −81 = 21 − 3n + 3 → −81 = 24 − 3n
• −105 = −3n → n = 35.

So −81 is the 35th term. Because n = 35 is a positive whole number, the value genuinely belongs to the AP.

A quick follow-up you should be able to answer: is any term of this AP equal to 0? Set 21 + (n − 1)(−3) = 0, which gives 3(n − 1) = 21, so n = 8. Yes — the 8th term is 0.

🎬 Watch (60-sec Short): Which term of this AP is exactly 0? 🤔

The key check every time: after solving, n must be a positive whole number. If it is not, the value is not a term (more on that below).

How to find an AP when two terms are given

Sometimes you are not told a and d directly — instead you are given two terms and asked to find the whole AP. Each given term becomes one equation, and two equations let you solve for the two unknowns a and d.

Method: write each given term using aₙ = a + (n − 1)d, then solve the two equations together (subtracting one from the other removes a and gives d).

Worked example. Determine the AP whose 3rd term is 5 and 7th term is 9.

• 3rd term: a + (3 − 1)d = a + 2d = 5 …(1)
• 7th term: a + (7 − 1)d = a + 6d = 9 …(2)
• Subtract (1) from (2): 4d = 4 → d = 1.
• Put d = 1 into (1): a + 2 = 5 → a = 3.

So the AP is 3, 4, 5, 6, 7, …. This is exactly the two-equations-two-unknowns idea you meet in the Pair of Linear Equations chapter, reused here. When the numbers get messier — awkward fractions, or the two terms far apart — that is where working through the fuller example set inside examfront, with personalised help where you slip, pays off.

How to check whether a number is a term of an AP

To test whether some number belongs to an AP, treat it as aₙ and solve for n. The AP only contains it if that position n turns out to be a positive whole number.

Method: set the number equal to aₙ in aₙ = a + (n − 1)d, solve for n, and check whether n is a positive integer.

Worked example. Is 301 a term of the AP 5, 11, 17, 23, …?

• a = 5, d = 6. Set 301 = 5 + (n − 1)×6.
• 301 = 5 + 6n − 6 → 301 = 6n − 1 → 6n = 302 → n = 302 ⁄ 6 = 50.33….
• n is not a whole number, so 301 is not a term of this AP.

If n had come out as, say, 51, the answer would be “yes, it is the 51st term.” The single deciding rule is: whole-number position → it’s a term; fraction or negative → it isn’t.

🎬 Watch (60-sec Short): Is 100 really a term of this AP? 🤔

A close cousin — counting terms. The same formula tells you how many terms an AP has when you know its last term. For example, how many two-digit numbers are divisible by 3? These are 12, 15, 18, …, 99, an AP with a = 12, d = 3 and last term l = 99. Using l = a + (n − 1)d: 99 = 12 + (n − 1)×3, so 87 = 3(n − 1), giving n − 1 = 29 and n = 30. There are 30 such numbers.

Where this leads next

Once you can find any term with aₙ = a + (n − 1)d, the natural next tool is the sum of the first n terms — a formula that adds up many terms at once (very handy for “total savings,” “total logs stacked,” or “total distance” problems). That builds straight on the nth term you have just learned.

We have kept this guide to the core of the nth-term topic and its most common question types. Harder mixed problems — such as comparing two different APs, “which term is so-much-more than another term,” middle-term puzzles, and full timed practice — are where you level up next. Work through the complete graded set inside examfront, and let Chapter Quizzes and Progress Tracking show you exactly which nth-term skills are exam-ready and which need one more round. For the full chapter map, see the Arithmetic Progressions chapter guide.

Key Takeaways

• The nth term (general term) of an AP is aₙ = a + (n − 1)d — first term plus the common difference added (n − 1) times.
• Multiply d by (n − 1), not n, because reaching the nth term takes (n − 1) steps from the first term.
• To find which term equals a value, set that value as aₙ and solve for n.
• To find an AP from two given terms, form two equations and solve for a and d together.
• A number is a term of an AP only if solving for n gives a positive whole number.

Quick Facts

• nth term formula: aₙ = a + (n − 1)d.
• aₙ: the nth term (the term you want). a: first term. d: common difference. n: position (a positive integer).
• Last term: l = a + (n − 1)d, where n is the total number of terms.
• Find a term: substitute n. Find which term: substitute the value, solve for n.
• Membership test: the number is a term ⇔ n is a positive whole number.
• Two-terms case: two equations → solve for a and d.
• Chapter: Arithmetic Progressions (Ch. 5) · Class: 10 · Subject: Maths · Board: CBSE.

Common Mistakes

1. Multiplying d by n instead of (n − 1). Writing a₁₀ = a + 10d instead of a + 9d is the most common slip. Remember the first term needs zero steps, so the nth term needs (n − 1) steps: aₙ = a + (n − 1)d.
2. Getting the sign of d wrong in a decreasing AP. For 21, 18, 15, …, d = 18 − 21 = −3, not +3. Always subtract the earlier term from the later one, then keep that negative sign inside the formula.
3. Accepting a fractional or negative n as a “term.” If solving for n gives 50.33 or −4, the value is not a term. A position must be a positive whole number — always check this before answering.
4. Confusing “find the term” with “find which term.” “Find the 12th term” means substitute n = 12 and compute aₙ. “Which term is 37?” means set aₙ = 37 and solve for n. Read which one the question wants.
5. Solving the two-terms case with only one equation. Two unknowns (a and d) need two equations. Write both given terms with the formula, then solve them together — don’t guess d from a single term.

FAQ

Q. What is the nth term of an AP? The nth term of an AP (also called the general term) is the term in the nth position of the list. It is given by aₙ = a + (n − 1)d, where a is the first term, d is the common difference and n is the position. For example, in 2, 7, 12, … the 10th term is a₁₀ = 2 + (10 − 1)×5 = 47.

Q. What is the formula for the nth term of an arithmetic progression? The formula is aₙ = a + (n − 1)d. Here aₙ is the term you want, a is the first term, d is the common difference, and n is the position. You multiply d by (n − 1), not by n, because reaching the first term takes zero steps.

Q. How do I find which term of an AP is equal to a given number? Put the known value into aₙ = a + (n − 1)d and solve for n. For 21, 18, 15, … to find which term is −81: −81 = 21 + (n − 1)(−3) gives n = 35, so −81 is the 35th term. If n is a fraction or non-positive, the value is not a term.

Q. How do you find an AP if only two terms are given? Write each given term using aₙ = a + (n − 1)d to get two equations in a and d, then solve them together. If the 3rd term is 5 and the 7th is 9, then a + 2d = 5 and a + 6d = 9; subtracting gives 4d = 4, so d = 1 and a = 3, and the AP is 3, 4, 5, 6, 7, ….

Q. How do I check whether a number is a term of an AP? Substitute the number as aₙ in aₙ = a + (n − 1)d and solve for n. If n is a positive whole number, it is a term; if n is a fraction or negative, it is not. For 5, 11, 17, 23, …, testing 301 gives n = 50.33, so 301 is not a term.

Related Concepts

• What Is an Arithmetic Progression? (Class 10): the first term, common difference and general form you need before using the nth-term formula.
• Arithmetic Progressions — Chapter Guide (Class 10): the full chapter roadmap, including the sum-of-n-terms formula that comes next.
• Class 10 Maths — All Formulas: the quick-reference sheet where aₙ = a + (n − 1)d lives.
• Pair of Linear Equations (Class 10): the two-equations method you use to find a and d from two given terms.

Ready to make the nth-term formula automatic? Practise a full graded set on examfront’s Topic Practice, and let Mistake Identification and Progress Tracking show you exactly where your (n − 1) slips and sign errors happen. Start on examfront →