Arithmetic Progressions Chapter 5 | Mathematics

Complete NCERT notes, nth term formula, sum of n terms, properties, selection of terms in AP, word problems, and key points for CBSE Class 10 Board Exam 2025-26.

📖 Introduction 🔢 General Form 📏 nth Term ➕ Sum of n Terms ⚡ Properties 🎯 Selection of Terms 📐 All Formulas 📝 Examples ⭐ Key Points

📖 Introduction to Arithmetic Progression (AP)

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

📌 Examples of AP

2, 4, 6, 8, 10,... (d = 2)

10, 7, 4, 1, -2,... (d = -3)

5, 5, 5, 5,... (d = 0)

💡 The common difference can be positive, negative, or zero.

🔢 General Form of an AP

📌 General Term Representation

a, a+d, a+2d, a+3d, ...

a = first term
d = common difference
n = number of terms
aₙ = nth term

📏 nth Term of an AP

📌 Formula

aₙ = a + (n - 1)d

📝 Example

Find the 10th term of AP: 5, 9, 13, 17,...
a = 5, d = 4, n = 10
a₁₀ = 5 + (10-1)×4 = 5 + 36 = 41

💡 Also: If aₘ and aₙ are given, d = (aₘ - aₙ)/(m - n)

➕ Sum of First n Terms of an AP

📌 Two Formulas for Sum Sₙ

Sₙ = n/2 [2a + (n-1)d]

Sₙ = n/2 (a + l) [where l = last term = aₙ]

📝 Example

Find sum of first 20 terms of AP: 2, 7, 12, 17,...
a = 2, d = 5, n = 20
S₂₀ = 20/2 [2×2 + (20-1)×5] = 10 [4 + 95] = 10 × 99 = 990

⚡ Important Properties of AP

If a constant is added to each term, the resulting sequence is also an AP with same d.
If each term is multiplied by a constant, the resulting sequence is also an AP with d multiplied by that constant.
Three terms in AP can be taken as: a-d, a, a+d
Four terms in AP: a-3d, a-d, a+d, a+3d
In an AP, the sum of terms equidistant from beginning and end is constant and equals sum of first and last term.

🎯 Smart Selection of Terms in AP

📌 3 Terms

Take as: a-d, a, a+d

Sum = 3a

📌 4 Terms

Take as: a-3d, a-d, a+d, a+3d

Sum = 4a

📌 5 Terms

Take as: a-2d, a-d, a, a+d, a+2d

Sum = 5a

📌 2 Terms

Take as: a, a+d

💡 This trick simplifies solving problems where sum and product of terms are given.

📐 All Important Formulas

1️⃣ nth Term

aₙ = a + (n-1)d

2️⃣ Sum of n Terms (using a,d)

Sₙ = n/2[2a + (n-1)d]

3️⃣ Sum of n Terms (using first & last)

Sₙ = n/2(a + l)

4️⃣ Common Difference

d = aₙ - aₙ₋₁

5️⃣ Number of Terms

n = (aₙ - a)/d + 1

6️⃣ nth term from end

aₙ' = l - (n-1)d

📝 NCERT Solved Examples (In-depth)

Example 1: Find nth term

Which term of AP: 3, 8, 13, 18,... is 78?
a = 3, d = 5, aₙ = 78
78 = 3 + (n-1)×5 → 75 = (n-1)×5 → n-1 = 15 → n = 16
∴ 78 is the 16th term.

Example 2: Sum of n terms

Find sum of first 24 terms of AP: 5, 8, 11, 14,...
a = 5, d = 3, n = 24
S₂₄ = 24/2 [2×5 + (24-1)×3] = 12 [10 + 69] = 12 × 79 = 948

Example 3: Word Problem (Installments)

A man saved ₹32 in first week, ₹40 in second week, ₹48 in third week,... How much does he save in 20 weeks?
AP: 32, 40, 48,... a=32, d=8, n=20
S₂₀ = 20/2[2×32 + 19×8] = 10[64 + 152] = 10×216 = ₹2160

Example 4: Selection of terms

Find three numbers in AP whose sum is 15 and product is 80.
Let terms: a-d, a, a+d
Sum = 3a = 15 → a = 5
Product = (5-d)×5×(5+d) = 80 → (25 - d²)×5 = 80 → 25 - d² = 16 → d² = 9 → d = ±3
Numbers: 2, 5, 8 or 8, 5, 2

Example 5: Sum of n natural numbers

Find sum of first 100 natural numbers.
AP: 1, 2, 3,...,100 → a=1, d=1, n=100
S₁₀₀ = 100/2(1+100) = 50×101 = 5050

⭐ Key Points for Board Exam (Deep Research)

AP is a sequence where difference between consecutive terms is constant.
nth term formula: aₙ = a + (n-1)d
Sum of n terms: Sₙ = n/2[2a + (n-1)d] = n/2(a + l)
If a, b, c are in AP → 2b = a + c
Three numbers in AP: a-d, a, a+d
Four numbers in AP: a-3d, a-d, a+d, a+3d
Sum of first n natural numbers: n(n+1)/2
Sum of first n odd numbers: n²
Sum of first n even numbers: n(n+1)
If Sₙ is given, term aₙ = Sₙ - Sₙ₋₁

📌 Quick Reference Card

Situation	Formula
nth term given a and d	aₙ = a + (n-1)d
Given aₘ and aₙ, find d	d = (aₘ - aₙ)/(m-n)
Sum of n terms	Sₙ = n/2[2a + (n-1)d]
Sum using first & last term	Sₙ = n/2(a + l)
Number of terms	n = (aₙ - a)/d + 1
Three terms in AP	a-d, a, a+d
If a, b, c in AP	2b = a + c

📌 Important Summation Results

Sum of first n natural numbers: 1+2+...+n = n(n+1)/2

Sum of first n odd numbers: 1+3+5+...+(2n-1) = n²

Sum of first n even numbers: 2+4+...+2n = n(n+1)

Arithmetic mean: A = (a+b)/2

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