Arithmetic Progression

Arithmetic progression Definition

An Arithmetic Progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This fixed value is called the common difference.

Example:

2, 5, 8, 11, 14...
Here, each number increases by 3. So, it’s an AP with common difference = 3.

Table of contents 

• Standard Notation in AP

• First Term of an AP

• Common Difference (d)

• General Form of an Arithmetic Progression

• Finding the nth Term of an AP

• Types of Arithmetic Progressions

• Use of the General Term Formula

• Sum of First n Terms (Sₙ)

• List of AP Formulas

• Questions and Step-by-Step Solutions

• Practice Problems (Problems to Solve)

• Related Articles

• Conclusion & Real-Life Relevance

• Frequently Asked Questions (FAQs)

Notation

Let’s understand how an arithmetic progression is generally written using symbols:

• First Term = a

• Common Difference = d

• nᵗʰ Term = Tₙ

• Number of Terms = n

• Sum of First n Terms = Sₙ
  A typical AP looks like:
  a, a + d, a + 2d, a + 3d, ...

First Term

The first term of an AP is simply the number that starts the sequence. It’s usually denoted by a.

Example:

In the sequence: 7, 10, 13, 16…
Here, the first term a = 7

Common Difference

The common difference (d) is the fixed amount added (or subtracted) to get from one term to the next.

Formula:
d = Second Term - First Term

Example:

In 7, 10, 13, 16…
d = 10 - 7 = 3

The difference is constant, so it’s an arithmetic progression.

General Form of an Arithmetic Progression

The general form of an AP is:
a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d

Here:

• a = first term

• d = common difference

• n = number of terms

• Tₙ = nᵗʰ term = a + (n - 1)d
  This format helps us find any term in the series without listing all the terms.

Finding the nth Term of an AP

The nth term of an arithmetic progression tells you the value of the term at position n in the sequence.

 Formula:

Tₙ = a + (n - 1)d

Example:

Find the 5th term of the AP: 4, 7, 10, 13…

Here, a = 4, d = 3, n = 5
T₅ = 4 + (5 - 1) × 3 = 4 + 12 = 16

Types of Arithmetic Progression

1. Increasing AP

When the common difference (d) is positive, each term gets bigger.
  Example: 2, 4, 6, 8…

2. Decreasing AP

When the common difference (d) is negative, each term gets smaller.
  Example: 20, 18, 16, 14…

3. Constant AP

When the common difference is 0, all terms are equal.
  Example: 5, 5, 5, 5…

Use of AP Formula for General Term

What is the General Term in an AP?
In an Arithmetic Progression (AP), each term increases (or decreases) by a fixed number. The general term helps you find any term in the sequence without writing all the previous ones.

 General Term Formula of an AP:

an=a+(n−1)da_n = a + (n - 1)d

Where:

• ana_n = the n-th term

• aa = first term of the AP

• dd = common difference

• nn = term number

 Why is this formula useful?

• Saves time: No need to write every term.

• Helps in solving word problems.

• Useful in real life - like calculating total payments or distances.
  
  

Example:

Let’s say an AP starts with 4 and increases by 3.
What is the 10th term?

Use the formula:

an=a+(n−1)d=4+(10−1)×3=4+27=31a_n = a + (n - 1)d = 4 + (10 - 1) \times 3 = 4 + 27 = 31

So, the 10th term is 31.

 Remember:

• If the common difference dd is negative, the terms will decrease.

• This formula works for any term - 5th, 50th, or even the 1000th!

Sum of n Terms of an AP (Sₙ)

To find the total of the first n terms of an AP, we use the formula:

Formula 1 (using d):

Sₙ = n/2 [2a + (n - 1)d]

Formula 2 (using first and last term):

Sₙ = n/2 × (first term + last term)

Example:

Find the sum of the first 5 terms of 3, 6, 9, 12, 15

a = 3, d = 3, n = 5
S₅ = 5/2 [2 × 3 + (5 - 1) × 3] = 5/2 [6 + 12] = 5/2 × 18 = 45

Arithmetic Progression Formula

Keep this list handy for solving AP problems quickly!

 Questions and Solutions

 Q1: Find the 10th term of the AP: 7, 11, 15, ...

Solution:
a = 7, d = 4, n = 10
T₁₀ = 7 + (10 - 1) × 4 = 7 + 36 = 43

 Q2: Find the sum of the first 8 terms of AP: 2, 5, 8, ...

Solution:
a = 2, d = 3, n = 8
S₈ = 8/2 × [2 × 2 + (8 - 1) × 3] = 4 × [4 + 21] = 4 × 25 = 100

 Q3: Is 55 a term in the AP: 3, 7, 11, 15…?

Solution:
a = 3, d = 4
Use Tₙ = a + (n - 1)d
Set 55 = 3 + (n - 1) × 4
⇒ 55 - 3 = 4(n - 1) ⇒ 52 = 4(n - 1) ⇒ (n - 1) = 13 ⇒ n = 14

Yes, 55 is the 14th term.

 Problems to Solve (Practice Section)

Try these questions on your own:

1. Find the 15th term of the AP: 10, 14, 18, ...

2. The 4th term of an AP is 20, and the 10th term is 50. Find a and d.

3. The sum of the first 12 terms of an AP is 420. If the first term is 10, find the common difference.

4. Find the sum of all two-digit numbers that are divisible by 5.

5. If the 7th term of an AP is 22 and the 13th term is 40, find the 20th term.

Conclusion : 

Arithmetic Progression is one of the most useful and easy-to-understand topics in algebra and number patterns. Once you master the basic formulas and practice enough problems, you’ll be able to apply this concept not just in exams, but in real-life situations like budgeting, planning savings, or even organizing events that follow a sequence.

Related Articles 

Addition and Subtraction - Addition and Subtraction to help others strengthen their foundation in math.

Arithmetic Mean Formula -  Arithmetic Mean Formula with your friends and classmates.

Algebra Formula  -  Algebra Formulas to help others conquer equations with ease.

 FAQs 

 Q1: What is the main difference between AP and GP?

Answer:
In an AP, the difference between terms is constant.
In a GP (Geometric Progression), the ratio between terms is constant.

 Q2: Can an AP have negative numbers?

Yes, if the common difference is negative, the AP will have decreasing terms, including negative numbers.

 Q3: Is 0 allowed in an AP?

Yes, zero can be a term or even the first term of an AP. For example: 0, 2, 4, 6..

 Q4: How do I know if a number is in a given AP?

Use the nth term formula:
Tₙ = a + (n - 1)d
Plug in the number and check if you get a whole number for n.

 Q5: What is the sum of an infinite AP?

An infinite AP does not have a finite sum unless all terms are zero. That’s why we only find the sum of a fixed number of terms.

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