Class 10 Notes - Arithmetic Progressions

Download PDF Worksheets
Start a Practice Test

Introduction

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is called the common difference, denoted by d.

1. Definitions

  • Term: Each number in the sequence.
  • Common Difference (d): d = an - an-1
  • General Form: a, a + d, a + 2d, a + 3d, …

2. Types of AP

  • Increasing AP: d > 0
  • Decreasing AP: d < 0
  • Constant AP: d = 0

3. nth Term of an AP

an = a + (n - 1)d

4. Sum of First n Terms of an AP

Sn = (n / 2)[2a + (n - 1)d]
or
Sn = (n / 2)(a + l)

5. Solving Problems

  • To find nth term: use an = a + (n - 1)d
  • To find number of terms: n = ((an - a) / d) + 1
  • To find sum: use the formula above

6. Important Results

  • Last term: l = a + (n - 1)d
  • Average of AP: (First Term + Last Term)/2

7. Word Problems

Example 1

5th term = 23, 9th term = 43

Let a = first term, d = common difference

a + 4d = 23
a + 8d = 43

Solving: d = 5, a = 3

Example 2

Find S20 for a = 7, d = 3

S20 = 10(2a + 19d) = 10(14 + 57) = 710

8. Special AP Types

  • Negative d: e.g., 50, 45, 40,…
  • Zero d: e.g., 6, 6, 6,…

9. Miscellaneous Examples

Q1: Which term is 88 in 3, 8, 13, …?

a = 3, d = 5 → 88 = a + (n - 1)d → n = 18

Q2: Sn = 5n² + 3n, find AP

S1 = 8, S2 = 26 → a2 = 18, S3 = 54 → a3 = 28

AP = 8, 18, 28,… → a = 8, d = 10

10. Graphical Representation

Points plotted as (term number, term value) lie on a straight line.

11. Mistakes to Avoid

  • Mixing nth term and sum formula
  • Missing +1 while calculating n
  • Incorrect substitution of variables

12. Summary Table

QuantityFormula
nth Terman = a + (n - 1)d
Sum of n TermsSn = n/2 [2a + (n - 1)d]
Common Differenced = a2 - a1
Last Terml = a + (n - 1)d
Number of Termsn = ((l - a)/d) + 1

13. Real-life Applications

  • Loan payments, salary increments
  • Stair design, bricks/tiles arrangement
  • Seating patterns, prize distribution

Conclusion

Arithmetic Progressions help us understand and solve problems involving regularly spaced numbers. Mastery of formulas and concept application is essential for success in board exams and real-world math problems.