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Arithmetic Progressions Class 10 Notes

Arithmetic Progressions Class 10 Notes

Arithmetic Progressions (AP) is a foundational chapter in CBSE Class 10 Maths that deals with sequences where the difference between consecutive terms is constant. These notes have been prepared by Lakshya Institution, Gurugram for board exam success with clear concepts, formulas, examples, and practice questions. :contentReference[oaicite:0]{index=0}

What is an Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence in which the difference between any two consecutive terms is always the same. For example: 3, 7, 11, 15, 19, … has a common difference of 4. :contentReference[oaicite:1]{index=1}

General Form of an AP

a, a + d, a + 2d, a + 3d, … Where a is the first term and d is the common difference. :contentReference[oaicite:2]{index=2}

Common Difference (d)

The constant difference between consecutive terms is called the common difference and is denoted by d. d = a₂ − a₁ = a₃ − a₂ = constant. :contentReference[oaicite:3]{index=3}

Nth Term of an AP

The nth term (aₙ) of an AP is given by: aₙ = a + (n − 1)d Where: a = first term, d = common difference, n = term number. :contentReference[oaicite:4]{index=4}

Sum of First n Terms

The sum (Sₙ) of the first n terms of an AP is given by: • Sₙ = n/2 [2a + (n − 1)d] • Sₙ = n/2 [a + l] Where l is the last term of the sequence. :contentReference[oaicite:5]{index=5}

Solved Examples (Board Pattern)

Example 1

Find the 20th term of the AP: 3, 6, 9, 12, …

Solution:
Here a = 3, d = 3
aₙ = a + (n − 1)d
a₂₀ = 3 + (20 − 1)×3 = 3 + 57 = 60

Example 2

Find the sum of first 15 terms of the AP: 5, 8, 11, 14, …

Solution:
a = 5, d = 3, n = 15
S₁₅ = n/2 [2a + (n − 1)d]
= 15/2 [10 + 14×3] = 15/2 [10 + 42] = 15/2 × 52 = 15 × 26 = 390

Previous Year Questions (PYQ)

PYQ 1 (CBSE Board)

Find the 12 terms AP if the first term is 7 and common difference is 4.

Solution:
a = 7, d = 4
AP: 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51

PYQ 2 (CBSE Board)

Find the sum of first 10 natural numbers using AP formula.

Solution:
Using Sₙ = n(n + 1)/2
Here n = 10
S₁₀ = 10×11/2 = 55

Important Practice Questions

  1. Find the 25th term of 8, 12, 16, 20, …
  2. Find the sum of first 20 terms of 4, 9, 14, 19, …
  3. Find whether 103 is a term of the AP 5, 10, 15, 20, …
  4. The sum of first n terms of AP is 210 and first term is 5, d = 3. Find n.
  5. If the nth term of an AP is 31 for a = 2 and d = 3, find n.

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