What Is an Arithmetic Progression? Introduction for Class 10 (AP)

In one line: an arithmetic progression (AP) is a list of numbers in which each term is made by adding the same fixed number to the term before it — like 5, 8, 11, 14, … where you keep adding 3.

That fixed number you keep adding is called the common difference, written as d. The very first number in the list is called the first term, written as a. So in 5, 8, 11, 14, …, the first term is a = 5 and the common difference is d = 3. This is the whole idea behind Section 5.1 of the Class 10 chapter Arithmetic Progressions.

Why does this matter? Patterns like this appear everywhere — a fixed monthly salary hike, rungs of a ladder getting shorter by the same amount, or money added to a savings box in equal steps. Once you can spot that a list “goes up (or down) by the same amount each time,” you can describe the entire list with just two numbers, a and d. The rest of this guide shows you how to find a and d, how to write an AP in its general form, and how to check whether any given list is an AP.

🎬 Watch (60-sec Short): Can you spot the common difference? ⚡

What is an arithmetic progression? (definition)

An arithmetic progression (AP) is a list of numbers in which each term after the first is obtained by adding a fixed number to the preceding term. Each number in the list is called a term.

That fixed number is the common difference (d). It is the “step size” of the AP. A key point students often miss: the common difference can be positive (the list grows), negative (the list shrinks), or zero (the list repeats the same number).

Here are three real-life APs from everyday situations:

• A starting salary of ₹8000 with a yearly raise of ₹500 gives 8000, 8500, 9000, … → here d = 500 (positive).
• A ladder whose rungs get 2 cm shorter from bottom to top gives 45, 43, 41, 39, … → here d = −2 (negative).
• ₹100 put in a money box on the first birthday, increased by ₹50 each year, gives 100, 150, 200, 250, … → here d = 50 (positive).

In every case the same number is added each time — that is exactly what makes each list an AP.

First term and common difference: a and d

To describe any AP completely, you only need two pieces of information:

• First term (a): the very first number in the list.
• Common difference (d): the fixed number added to each term to reach the next one.

How to find d — the rule (with a worked example). The common difference is found by subtracting any term from the term that immediately follows it:

d = a₍ₖ₊₁₎ − aₖ — the (k+1)th term minus the kth term.

In plain words: pick any term, subtract it from the next term, and you get d. You do not have to check every pair — for a genuine AP, one subtraction is enough (though checking two is a good safety habit).

Worked example. For the list 7, 11, 15, 19, …:

• First term: a = 7.
• Common difference: d = 11 − 7 = 4 (and 15 − 11 = 4, 19 − 15 = 4 confirm it).

So this is an AP with a = 7 and d = 4.

Important order rule: always subtract the earlier term from the later term, even when the numbers are going down. For 21, 18, 15, …, the common difference is d = 18 − 21 = −3, not +3. Getting this sign right is one of the most common exam slip-ups — the kind of small error that examfront’s Mistake Identification flags for you the moment it happens.

The general form of an AP

Once you know a and d, you can build the whole progression. Every AP can be written in this general form:

a, a + d, a + 2d, a + 3d, …

where a is the first term and d is the common difference. Reading it off: the 2nd term is a + d, the 3rd term is a + 2d, and so on — each term adds one more d.

Worked example. If a = 6 and d = 3, the AP is:

6, (6 + 3), (6 + 2×3), (6 + 3×3), … = 6, 9, 12, 15, …

And if a = 6 but d = −3, the same rule gives 6, 3, 0, −3, …. This is why a and d together are enough to reconstruct any AP — a neat idea you can drill to fluency with Topic Practice on examfront.

Finite and infinite APs

APs come in two types, depending on whether the list ever ends:

• A finite AP has a last term. Example: the heights of students in a queue, 147, 148, 149, …, 157 — it stops at 157.
• An infinite AP has no last term and continues forever. Example: 2, 5, 8, 11, … with the “…” meaning it never stops.

The definition, the first term, and the common difference all work exactly the same way for both types — the only difference is whether there is a final term. The last term of a finite AP is often written as l.

How to check whether a list is an AP

This is one of the most-asked exam questions from this section, so here is a clean method.

The test: find the difference between each pair of consecutive terms. If all the differences are equal, the list is an AP (and that equal value is d). If even one difference is different, it is not an AP.

Worked example 1 (it is an AP). Is 4, 10, 16, 22, … an AP?

• 10 − 4 = 6, 16 − 10 = 6, 22 − 16 = 6.
• All differences equal 6 → yes, it is an AP with d = 6. The next term is 22 + 6 = 28.

Worked example 2 (it is not an AP). Is 1, 4, 9, 16, … an AP?

• 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7.
• The differences 3, 5, 7 are not equal → no, it is not an AP (these are perfect squares, a different pattern).

🎬 Watch (60-sec Short): Which list is NOT an arithmetic progression? 🤔

Watch out for disguised APs. Some lists look scary but still pass the test. For example, √3, √12, √27, √48 simplifies to √3, 2√3, 3√3, 4√3 — the differences are all √3, so it is an AP. Simplify first, then apply the test. The trickier “spot-the-AP” cases like surds and algebraic terms are exactly where extra practice pays off — work through a full graded set inside examfront to make these automatic.

🎬 Watch (60-sec Short): Can these surds form an arithmetic progression? 🤔

Where this leads next

Section 5.1 gives you the foundation: what an AP is, how to find a and d, and how to recognise one. From here, Chapter 5 builds two powerful tools on top of this idea — the nth term formula (to jump straight to, say, the 100th term without listing them all) and the sum of the first n terms formula (to add up many terms quickly). Those are the higher-marks parts of the chapter.

We have kept this introduction focused on the core of Section 5.1. For the full chapter roadmap — including the nth-term and sum formulas, harder identification problems, and complete practice — see the Arithmetic Progressions chapter guide and continue on examfront, where Chapter Quizzes and Progress Tracking show you exactly which AP skills are exam-ready and which need another round.

Key Takeaways

• An arithmetic progression (AP) is a list of numbers where each term is the previous term plus a fixed number.
• That fixed number is the common difference (d); the first number is the first term (a).
• Find d by subtracting a term from the next term: d = a₍ₖ₊₁₎ − aₖ — it can be positive, negative or zero.
• The general form of an AP is a, a + d, a + 2d, a + 3d, … — so a and d describe the whole list.
• A list is an AP only if every consecutive difference is the same; a finite AP has a last term, an infinite AP does not.

Quick Facts

• Arithmetic progression (AP): a list where each term = previous term + a fixed number.
• Common difference (d): the fixed number added each time; d = a₍ₖ₊₁₎ − aₖ.
• First term (a): the first number of the AP.
• General form: a, a + d, a + 2d, a + 3d, ….
• d can be: positive (increasing), negative (decreasing), or zero (constant).
• AP test: all consecutive differences must be equal.
• Finite AP: has a last term (l). Infinite AP: no last term.
• Chapter: Arithmetic Progressions (Ch. 5) · Class: 10 · Subject: Maths · Board: CBSE.

Common Mistakes

1. Getting the sign of d wrong. For a decreasing list like 21, 18, 15, …, students write d = +3. Always subtract the earlier term from the later one: d = 18 − 21 = −3.
2. Assuming any pattern is an AP. Squares (1, 4, 9, 16, …) and doubling (2, 4, 8, 16, …) are patterns but not APs, because the differences are not equal. Always run the difference test.
3. Checking only one pair of terms. Finding that one difference “looks right” and stopping can hide a trap. Check at least two consecutive differences before deciding.
4. Confusing “term” with “common difference.” The terms are the numbers in the list; d is the gap between them. Mixing these up leads to wrong answers in fill-in-the-blank AP questions.
5. Panicking at fractions or surds. Lists like 3/2, 1/2, −1/2, … or √3, √12, √27, … are still APs. Simplify carefully; here d = 1/2 − 3/2 = −1 and d = √3 respectively.

FAQ

Q. What is an arithmetic progression in Class 10 Maths? An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding the same fixed number to the term before it, except the first term. That fixed number is the common difference. For example, 5, 8, 11, 14, … is an AP because you keep adding 3.

Q. What is the common difference of an AP? The common difference (d) is the fixed number added to each term to get the next one. Find it by subtracting any term from the term right after it: d = a₍ₖ₊₁₎ − aₖ. It can be positive, negative or zero. For 5, 8, 11, 14, …, d = 8 − 5 = 3.

Q. What is the general form of an arithmetic progression? The general form is a, a + d, a + 2d, a + 3d, …, where a is the first term and d is the common difference. Just knowing a and d lets you write the entire AP. For a = 6 and d = 3, the AP is 6, 9, 12, 15, ….

Q. How do I check whether a list of numbers is an AP? Subtract each term from the one after it. If every difference is the same, the list is an AP and that value is the common difference. If even one difference is different, it is not an AP. For example, 1, 4, 9, 16, … is not an AP because 3, 5, 7 are not equal.

Q. What is the difference between a finite and an infinite AP? A finite AP has a last term (like 2, 5, 8, …, 20), while an infinite AP goes on forever with no last term (like 2, 5, 8, 11, …). The first term and common difference behave exactly the same way in both.

Related Concepts

• Arithmetic Progressions — Chapter Guide (Class 10): the full chapter roadmap, including the nth term and sum-of-n-terms formulas.
• Class 10 Maths — All Formulas: the quick-reference formula sheet where the AP formulas live.
• Real Numbers (Class 10): number sense and patterns that make spotting APs easier.
• Pair of Linear Equations (Class 10): the two-equation method you will use later to find a and d from two given terms.

Ready to make APs automatic? Practise a full graded set on examfront’s Topic Practice, and let Mistake Identification and Progress Tracking show you exactly where your sign slips and pattern traps happen. Start on examfront →