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Class 10 · Maths · Chapter 5 · NCERT Class 10 Mathematics

Arithmetic Progressions Class 10 Notes

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Chapter mind map

The whole chapter at a glance: the big idea, then each branch and what sits under it.

Arithmetic Progressions (AP)

A sequence where each term is obtained by adding a fixed number (common difference) to the preceding term.

  1. Fundamentals of AP

    Defining the structure, components, and identification of arithmetic sequences.

    • Real-world Patterns — Observed in sunflower petals, honeycombs, salary increments, and ladder rungs.
    • Core Components — Defined by first term 'a' and common difference 'd' (positive, negative, or zero).
    • Verification Method — A list is an AP if differences a2-a1, a3-a2, etc., are all equal to 'd'.
    • Sequence Types — Finite APs have a last term; infinite APs continue indefinitely without end.

  2. The nth Term (an)

    The general term formula used to find specific positions or values in a sequence.

    • General Form Formula — an = a + (n - 1)d, where n must be a positive integer (1, 2, 3...).
    • Solving for n — Used to count terms in a range, such as two-digit numbers divisible by 3.
    • Terms from the End — Found by reversing the AP (new a = l, new d = -d) or using total term count.
    • Last Term (l) — In a finite AP with 'm' terms, the last term am is often denoted as 'l'.

  3. Sum of First n Terms (Sn)

    Calculating the total of an AP using formulas derived from Gauss's method.

    • Standard Sum Formula — Sn = (n/2)[2a + (n - 1)d], involving four quantities: Sn, n, a, and d.
    • Simplified Sum Formula — Sn = (n/2)(a + l), useful when the first and last terms are known.
    • Finding an from Sn — The nth term is the difference between consecutive sums: an = Sn - Sn-1.
    • Quadratic Solutions — Solving for n may yield two valid answers if 'd' is negative and terms cancel.

  4. Advanced Properties

    Arithmetic mean and algebraic representations of sequences.

    • Arithmetic Mean — If a, b, c are in AP, then b = (a + c) / 2; b is the mean of a and c.
    • Missing Term Logic — Mean property helps find unknown values, like the middle of 2, [ ], 26.
    • Algebraic Expressions — If an = 3 + 2n, find 'a' and 'd' by substituting n = 1, 2 to solve for sums.
    • Divisibility Problems — Steps: Identify first term 'a', difference 'd', last term 'an', then solve for 'n'.

Chapter notes

This chapter explores the mathematical patterns where each term is obtained by adding a fixed number to the preceding term. It covers the general form of an AP, finding the nth term, and calculating the sum of the first n terms.

01 · Explore

Introduction to Patterns

In nature and daily life, we often encounter sequences of numbers that follow a specific rule or pattern.

Consider the petals of a sunflower, the holes of a honeycomb, or the spirals on a pine cone. These follow mathematical structures. In our daily lives, we see patterns in financial growth or physical measurements. For example, if a person starts a job with a monthly salary of 8000 and receives an annual increment of 500, their salary for successive years will be 8000, 8500, 9000, and so on.

Similarly, the rungs of a ladder might decrease uniformly in length from bottom to top. If the bottom rung is 45 cm and each subsequent rung is 2 cm shorter, the lengths are 45, 43, 41, 39, etc. In these examples, we observe that each succeeding term is obtained by adding a fixed number to the preceding term. This specific type of pattern is the focus of this chapter.

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In the sequence 100, 150, 200, 250, ..., what is the rule for finding the next term?

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NCERT reference: chapter PDF pages 1, 2, 3.

02 · Explore

Defining Arithmetic Progression (AP)

An Arithmetic Progression is a list of numbers where the difference between consecutive terms remains constant.

An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except for the first term. This fixed number is called the common difference, denoted by 'd'. The first term is typically denoted by 'a'.

The common difference 'd' can be positive, negative, or zero. If d is positive, the terms increase; if d is negative, the terms decrease; and if d is zero, all terms remain the same (e.g., 3, 3, 3, 3, ...).

To verify if a list of numbers is an AP, we calculate the difference between consecutive terms: a₂ - a₁, a₃ - a₂, a₄ - a₃, and so on. If all these differences are equal to the same value 'd', the list is an AP.

SequenceFirst Term (a)Common Difference (d)Type of AP
1, 2, 3, 4, ...11Infinite
100, 70, 40, 10, ...100-30Infinite
-3, -2, -1, 0-31Finite
3, 3, 3, 3, ...30Infinite

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Can the common difference of an AP be zero?

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NCERT reference: chapter PDF pages 3, 4, 5.

03 · Explore

General Form of an AP

The general form allows us to represent any AP using its first term and common difference.

If 'a' is the first term and 'd' is the common difference, the terms of the AP are written as: a, a + d, a + 2d, a + 3d, ... and so on. This is known as the general form of an Arithmetic Progression.

APs are classified into two types: Finite and Infinite. A finite AP has a limited number of terms and always has a last term (e.g., the heights of 10 students in a queue). An infinite AP does not have a last term and continues indefinitely (e.g., the set of all natural numbers).

To define an AP completely, we must know both the first term 'a' and the common difference 'd'. For instance, if a = 6 and d = 3, the AP is 6, 9, 12, 15, ...

Identifying a and d

Sequence: 3/2, 1/2, -1/2, -3/2, ...

The first term a = 3/2. To find d, subtract the first term from the second: d = 1/2 - 3/2 = -2/2 = -1. Thus, a = 1.5 and d = -1.

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Write the first four terms of an AP where a = -2 and d = 0.

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NCERT reference: chapter PDF pages 4, 5.

04 · Explore

The nth Term of an AP

The nth term formula allows us to find any specific term in a sequence without listing all previous terms.

Let a₁, a₂, a₃, ... be an AP with first term 'a' and common difference 'd'. We observe a pattern: the 2nd term is a + d, the 3rd term is a + 2d, and the 4th term is a + 3d. In each case, the coefficient of 'd' is one less than the position of the term.

Generalizing this, the nth term (aₙ) is given by the formula: aₙ = a + (n - 1)d. Here, 'n' represents the position of the term and must always be a positive integer.

The nth term is also called the general term of the AP. If an AP has 'm' terms, then aₘ represents the last term, often denoted by the letter 'l'.

Finding the 10th Term

AP: 2, 7, 12, ... find a₁₀

Here a = 2, d = 7 - 2 = 5, and n = 10. Using aₙ = a + (n - 1)d: a₁₀ = 2 + (10 - 1)5 = 2 + (9 × 5) = 2 + 45 = 47.

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If a = 21 and d = -3, which term is -81?

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NCERT reference: chapter PDF pages 8, 9, 10.

The rest of this chapter

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  1. Locked: 1. Applications of the nth Term
  2. Locked: 2. Sum of First n Terms
  3. Locked: 3. Complex Summation Problems
  4. Locked: 4. Arithmetic Mean

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