Introduction to Arithmetic Progression
Introduction of arithmetic progression.
Chapter 6 — Introduction to Progressions
Chapter 6, Progressions, is one of the most pattern-rich and interesting chapters in Class 10 Mathematics (CBSE, Telangana & AP Board). A progression is simply a sequence of numbers that follows a logical, predictable rule — each term is connected to the next by the same relationship. From salary increments to staircase steps, from bank interest to falling objects, progressions appear everywhere in real life.
This introduction lesson covers what a progression is, what an Arithmetic Progression (AP) is, how to identify one, what the common difference (d) means, and how to write the general form of an AP. These are the building blocks for the entire chapter — every formula for the n-th term and sum of an AP is built on these ideas.
What is a Progression?
A progression is a sequence of numbers where each term follows the previous one according to a fixed, consistent rule or pattern. Unlike a random list of numbers, in a progression you can always predict the next term once you know the rule. Each individual number in the progression is called a term.
The six examples from the lesson — all valid progressions, each with a different underlying rule:
| # | Sequence | Rule / Pattern | Type |
|---|---|---|---|
| 1 | 2, 4, 6, 8, … | Add 2 each time | AP (d = 2) |
| 2 | 1, 4, 9, 16, 25, … | Perfect squares (n²) | Sequence (not AP) |
| 3 | 1, 2, 4, 7, 11, … | Differences increase: +1, +2, +3, +4… | Sequence (not AP) |
| 4 | 3, 6, 12, 24, 48, … | Multiply by 2 each time | GP (r = 2) |
| 5 | 1/3, 1/7, 1/11, 1/15, … | Denominators form AP (+4 each time) | Sequence |
| 6 | 2/3, 5/7, 8/11, 11/15, … | Numerators AP (+3), Denominators AP (+4) | Sequence |
Three Main Types of Progressions in Class 10
d = +4
r = ×3
(perfect squares)
What is an Arithmetic Progression (AP)?
An Arithmetic Progression (AP) is a special sequence where each term is obtained by adding or subtracting the same fixed number to the term before it — starting from the second term. This fixed number is called the common difference, denoted by d.
The word "common" means the same difference occurs between every consecutive pair of terms. "Arithmetic" refers to the additive (constant difference) nature of the pattern — as opposed to a Geometric Progression which uses multiplication.
d = a₂ − a₁ = a₃ − a₂ = a₄ − a₃ = … = aₙ − aₙ₋₁The difference between any two consecutive terms is always the same constant d.
Five Examples of Arithmetic Progressions — Explained Step by Step
The lesson covers five distinct AP examples showing increasing, decreasing, fractional, negative, and decimal common differences. Understanding all five builds the intuition to identify any AP instantly.
This is the sequence of odd numbers starting from 3. It is one of the most commonly seen APs in board exam problems — "Find the 20th term of 3, 5, 7, 9, …" type questions appear almost every year.
A decreasing AP has a negative common difference. Students often think "subtraction means it's not an AP" — that's incorrect. Any constant difference, positive or negative, makes it an AP.
AP terms don't have to be whole numbers. As long as the difference between consecutive terms is constant, it is an AP — even with fractions.
When all terms are negative and decreasing, always verify d by subtracting a term from the next term (not the other way around): d = a₂ − a₁, not a₁ − a₂. Here −6 − (−2) = −4.
This AP is equivalent to multiplying the sequence 1, 2, 3, 4, 5, … by 0.5. Decimal APs are common in problems about measurements, temperatures, and financial data.
General Form of an Arithmetic Progression
If the first term of an AP is a and the common difference is d, then every subsequent term is obtained by adding d repeatedly. This gives us the general form of any AP:
Notice the pattern: the nth term always has (n−1) copies of d added to a. The 1st term has 0×d = 0 (no d added), the 2nd term has 1×d, the 3rd has 2×d, and so on. This observation is the foundation of the n-th term formula aₙ = a + (n−1)d, which you will use throughout the chapter.
aₙ = a + (n − 1)dwhere a = first term, d = common difference, n = term number
The Defining Property of an AP — Equal Consecutive Differences
For a sequence a₁, a₂, a₃, …, aₙ₋₁, aₙ to be an AP, one condition must hold throughout: the difference between any consecutive pair of terms must be the same constant d.
a₂ − a₁ = a₃ − a₂ = a₄ − a₃ = … = aₙ − aₙ₋₁ = d (constant)
This property is both the definition of an AP and the most practical way to check whether a given sequence is an AP — you simply subtract consecutive terms and check if the result is always the same number.
How to Check if a Sequence is an AP — Step-by-Step
2−1 = 1 | 4−2 = 2 | 7−4 = 3 | 11−7 = 4
The differences are 1, 2, 3, 4 — not equal. So this is NOT an AP even though it follows a pattern.
Summary — All Five AP Examples at a Glance
| # | Sequence | First Term (a) | Common Difference (d) | Direction | Type of d |
|---|---|---|---|---|---|
| 1 | 3, 5, 7, 9, 11, … | 3 | +2 | Increasing | Positive integer |
| 2 | 16, 12, 8, 4, … | 16 | −4 | Decreasing | Negative integer |
| 3 | 1/4, 1/2, 3/4, 1, 5/4, … | 1/4 | +1/4 | Increasing | Positive fraction |
| 4 | −2, −6, −10, −14, … | −2 | −4 | Decreasing | Negative integer |
| 5 | 0.5, 1, 1.5, 2, 2.5, … | 0.5 | +0.5 | Increasing | Positive decimal |
Common Mistakes to Avoid
- Computing d in the wrong direction: Always compute d = a₂ − a₁ (next term minus previous term), not a₁ − a₂. For 16, 12, 8, 4: d = 12 − 16 = −4, not +4.
- Assuming AP must be increasing: A decreasing sequence (like 16, 12, 8, 4) is a perfectly valid AP with a negative common difference. "Arithmetic" does not mean "growing."
- Confusing GP and AP: 3, 6, 12, 24 is a GP (multiply by 2), not an AP (differences are 3, 6, 12 — not constant). Always compute differences, not ratios, when checking for AP.
- Thinking d = 0 is not allowed: If d = 0, every term equals a — a valid (if boring) AP. For example: 5, 5, 5, 5, … is an AP with d = 0.
- Only checking one pair of consecutive terms: You must verify d is the same for ALL consecutive pairs, not just the first two. The sequence 2, 4, 6, 9 looks like an AP for the first three terms but fails at the 4th (9 − 6 = 3 ≠ 2).
What This Introduction Prepares You For
This introduction to progressions directly prepares you for every subsequent topic in Chapter 6. The concepts of first term (a) and common difference (d) are the two inputs needed for the Exercise 6.1 problems on identifying and writing APs. They also feed directly into the n-th term formula and the sum formula covered in Exercises 6.2 and 6.3.
At a broader level, the pattern-recognition skills developed here connect to Real Numbers (where number patterns appear in Euclid's algorithm) and lay conceptual groundwork for Geometric Progressions and series in higher classes. For Class 9 students building toward this chapter, related pattern work appears in Polynomials.