Arithmetic Progression

Arithmetic Progression (AP), also called Arithmetic Sequence, is a type of number pattern where the difference between two consecutive terms is always the same.

 Example: The natural numbers 1, 2, 3, 4, 5, … form an AP because each number increases by 1.
 Example: The even numbers 2, 4, 6, 8, … also form an AP because each number increases by 2.

We see such patterns everywhere in daily life - roll numbers in a class, days of the week, or months in a year. That’s why Arithmetic Progression is one of the most useful concepts in mathematics.

 

Table of Contents

• What is Arithmetic Progression?
• Notation
• First Term
• Common Difference
• General Form
• Nth Term
• Types of AP (Finite & Infinite)
• Sum of N Terms
• Formula List
• Solved Examples
• Practice Questions
• FAQs on Arithmetic Progression

What is Arithmetic Progression?

In mathematics, a progression is a sequence of numbers that follows a rule. There are mainly three types of progressions:

1. Arithmetic Progression (AP)

2. Geometric Progression (GP)

3. Harmonic Progression (HP)
   
   

Out of these, AP is the simplest and most commonly used.

Definition 1: An AP is a sequence of numbers where the difference between two consecutive terms is always constant.

Definition 2: In an AP, every term after the first is obtained by adding the same fixed number (called the common difference) to the previous term.

Example: 3, 6, 9, 12, … has a common difference of 3.

 

Notation in Arithmetic Progression

When we talk about an AP, we usually use:

• a = first term

• d = common difference

• an = nth term

• Sn = sum of first n terms
  
  

First Term of an AP

If the first term is a, then the sequence looks like:

a, a + d, a + 2d, a + 3d, …, a + (n – 1)d

Common Difference

If an AP is a1, a2, a3, …, then:

d = a2 – a1 = a3 – a2 = … = an – an-1

The value of d can be positive, negative, or zero.

General Form of an AP

nth Term of an AP

Formula:
an = a + (n – 1) × d

 Example: Find the 15th term of AP: 1, 2, 3, …

a = 1, d = 1, n = 15

a15 = 1 + (15 – 1) × 1 = 15

 

Types of Arithmetic Progression

1. Finite AP - An AP that has a limited number of terms.
     Example: 5, 10, 15, 20, 25

2. Infinite AP - An AP that continues endlessly without a last term.
     Example: 2, 4, 6, 8, 10, …
   
   

Sum of First n Terms of AP

The formula is:

Sn = n/2 [2a + (n – 1) × d]

If the last term l is known, then:

Sn = n/2 (a + l)

 Example: Find the sum of first 15 natural numbers.
a = 1, d = 1, n = 15

Sn = 15/2 [2(1) + (15 – 1)(1)]
= 15/2 [2 + 14] = 15 × 8 = 120

 

Formula List :

 

Solved Examples

Example 1: If a = 10, d = 5, and an = 95, find n.
Solution: 95 = 10 + (n – 1)5 → n = 18

Example 2: Find the 20th term of AP: 3, 5, 7, 9…
Solution: a = 3, d = 2, n = 20 → a20 = 41

Example 3: Find the sum of first 30 multiples of 4.
Solution: a = 4, d = 4, n = 30
S30 = 30/2 [2(4) + (30 – 1)(4)] = 1860

Practice Questions

1. Find the 10th term of 3, 1, 17, 24, …

2. If a = 2, d = 3, n = 90, find an and Sn.

3. The 7th term is 12 and the 10th term is 25. Find the 12th term.
   
   

Frequently Asked Questions on Arithmetic Progression

Q1. What is the general form of an AP?

Answer. a, a + d, a + 2d, …, a + (n – 1)d

Q2. Give one simple example of AP.

Answer. The sequence 5, 10, 15, 20, … is an AP with common difference 5.

Q3. How do you find the sum of AP?

Answer. Use the formula Sn = n/2 [2a + (n – 1)d].

Q4. What are the types of progressions?

Answer. Arithmetic Progression (AP), Geometric Progression (GP), Harmonic Progression (HP).

Q5. Where is AP used in real life?

Answer. AP is used in predicting patterns, like calculating salary hikes, cab arrival time at equal intervals, or arranging seats in rows.

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