Short Notes: Arithmetic Progressions

# Arithmetic Progressions Class 10 Notes Maths Chapter 5

 Table of contents What is a Sequence ? Arithmetic Progression (A.P.) nth term of an A.P. Sum of n terms of an A.P.

## What is a Sequence ?

A sequence is an arrangement of numbers in a definite order and according to some rule.
Example:

•  2, 4, 6, 8, 10, … is a sequence where each successive item is 2 greater than the preceding term.

• 1, 4, 9, 16, 25, … is a sequence where each term is the square of successive natural numbers.

### Terms

The various numbers occurring in a sequence are called ‘terms’. Since the order of a sequence is fixed, therefore the terms are known by the position they occupy in the sequence.
Example: If the sequence is defined as

Question for Short Notes: Arithmetic Progressions
Try yourself:In the sequence 3, 6, 12, 24, 48, ..., what is the rule or pattern that governs the sequence?

## Arithmetic Progression (A.P.)

An Arithmetic progression is a special case of a sequence, where the difference between a term and its preceding term is always constant, known as common difference, i.e., d. The arithmetic progression is abbreviated as A.P.

### General form of an A.P.

The general form of an A.P. is
∴ a, a + d, a + 2d,… For example, 1, 9, 11, 13.., Here the common difference is 2. Hence it is an A.P.

## nth term of an A.P.

In an A.P. with first term a and common difference d, the nth term (or the general term) is given by .

an = a + (n – 1)d.

where [a = first term, d = common difference, n = term number

For Example,

To find seventh term put n = 7
∴ a7 = a + (7 – 1)d or a7 = a + 6d

### Solved Examples

Example 1: Find the value of n, if a = 10, d = 5, an = 95.

Sol: Given, a = 10, d = 5, an = 95

From the formula of general term, we have:

an = a + (n − 1) × d

95 = 10 + (n − 1) × 5

(n − 1) × 5 = 95 – 10 = 85

(n − 1) = 85/ 5

(n − 1) = 17

n = 17 + 1

n = 18

Example 2: Find the 20th term for the given AP:3, 5, 7, 9, ……

Sol: Given,

3, 5, 7, 9, ……

a = 3, d = 5 – 3 = 2, n = 20

an = a + (n − 1) × d

a20 = 3 + (20 − 1) × 2

a20 = 3 + 38

⇒a20 = 41

Question for Short Notes: Arithmetic Progressions
Try yourself:What is the 20th term of the arithmetic progression (A.P.) 3, 5, 7, 9, ...?

## Sum of n terms of an A.P.

The sum of the first n terms of an A.P. is given by

where [a = first term, d = common difference, n = term number

ExampleFind the sum of the first 30 multiples of 4.

Sol: The first 30 multiples of 4 are: 4, 8, 12, ….., 120

Here, a = 4, n = 30, d = 4

We know,

S30 = n/2 [2a + (n − 1) × d]

S30 = 30/2[2 (4) + (30 − 1) × 4]

S30 = 15[8 + 116]

S30 = 1860

Note: If a, b, c are in A.P. then b =  and b is called the arithmetic mean of a and c.

The document Arithmetic Progressions Class 10 Notes Maths Chapter 5 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Arithmetic Progressions Class 10 Notes Maths Chapter 5

 1. What is an Arithmetic Progression (A.P.)?
Ans. An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant.
 2. How do you find the nth term of an Arithmetic Progression (A.P.)?
Ans. The formula to find the nth term of an A.P. is: $$a_n = a_1 + (n-1)d$$, where $$a_n$$ is the nth term, $$a_1$$ is the first term, $$n$$ is the term number, and $$d$$ is the common difference.
 3. What is the formula for the sum of n terms of an Arithmetic Progression (A.P.)?
Ans. The formula to find the sum of n terms of an A.P. is: $$S_n = \frac{n}{2} [2a_1 + (n-1)d]$$, where $$S_n$$ is the sum of n terms, $$a_1$$ is the first term, $$n$$ is the number of terms, and $$d$$ is the common difference.
 4. How can you identify if a given sequence is an Arithmetic Progression (A.P.)?
Ans. To identify if a sequence is an A.P., check if the difference between consecutive terms is constant. If the difference remains the same throughout, then it is an A.P.
 5. Can the common difference in an Arithmetic Progression (A.P.) be negative?
Ans. Yes, the common difference in an A.P. can be negative. This means that each term in the sequence decreases by the same value.

## Mathematics (Maths) Class 10

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