📘 Arithmetic Progressions Class 10 Notes (Chapter 5)

🔷 Introduction

Arithmetic Progression, also called AP, is an important topic in mathematics. In simple words, an arithmetic progression is a sequence of numbers where the difference between consecutive terms remains the same.

For example:

2, 5, 8, 11, 14…

Here, the common difference is 3.

Therefore, this sequence is an arithmetic progression.

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🔹 What is Arithmetic Progression?

A sequence is called an arithmetic progression if:$$a_2-a_1 = a_3-a_2 = a_4-a_3$$a2​−a1​=a3​−a2​=a4​−a3​

This fixed difference is called the Common Difference (d).

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🔹 Important Terms in AP

✔ First Term (a)

The first number of the sequence is called the first term.

Example:
In 3, 7, 11, 15…
First term = 3

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✔ Common Difference (d)

The difference between two consecutive terms is called the common difference.

Formula:$$d=a_n-a_{n-1}$$d=an​−an−1​

Example:

7−3 = 4

So, d = 4

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✔ Number of Terms (n)

The total number of terms is represented by n.

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🔷 General Form of AP

The general form of arithmetic progression is:$$a,\ a+d,\ a+2d,\ a+3d,\dots$$a, a+d, a+2d, a+3d,…

Thus, every term can be found easily.

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🔷 nth Term of AP

The nth term of an arithmetic progression is:$$a_n=a+(n-1)d$$an​=a+(n−1)d

Where:

• a = first term
• d = common difference
• n = number of terms

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🔹 Example of nth Term

Find the 10th term of:

2, 5, 8, 11…

Here:
a = 2
d = 3
n = 10

Using formula:$$a_n=a+(n-1)d$$an​=a+(n−1)d $$a_{10}=2+(10-1)\times3$$a10​=2+(10−1)×3 $$a_{10}=2+27$$a10​=2+27 $$a_{10}=29$$a10​=29

Hence, the 10th term is 29.

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🔷 Sum of n Terms of AP

The sum of first n terms is:$$S_n=\frac{n}{2}[2a+(n-1)d]$$Sn​=2n​[2a+(n−1)d]

Another important formula is:$$S_n=\frac{n}{2}(a+l)$$Sn​=2n​(a+l)

Where l is the last term.

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🔹 Example of Sum Formula

Find the sum of first 5 terms:

2, 5, 8, 11, 14…

Here:
a = 2
d = 3
n = 5

Using formula:$$S_n=\frac{n}{2}[2a+(n-1)d]$$Sn​=2n​[2a+(n−1)d] $$S_5=\frac{5}{2}[4+12]$$S5​=25​[4+12] $$S_5=\frac{5}{2}\times16$$S5​=25​×16 $$S_5=40$$S5​=40

Therefore, the sum is 40.

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🔷 Important Properties of AP

✔ If d > 0

The sequence increases.

✔ If d < 0

The sequence decreases.

✔ If d = 0

All terms become equal.

As a result, the commo📘 Arithmetic Progressions Class 10 Notes (Chapter 5)