Class 10 Maths Arithmetic Progressions Notes

5.1 Introduction to Arithmetic Progressions

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

For example:

• $2, 5, 8, 11, 14, \dots$2,5,8,11,14,… is an arithmetic progression with the common difference $d = 3$d=3.
• $20, 17, 14, 11, 8, \dots$20,17,14,11,8,… is also an AP, but with a common difference $d = -3$d=−3.

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

Where:

• $a$a is the first term,
• $d$d is the common difference,
• The $n$n-th term is represented as $a_n = a + (n – 1) \cdot d$an​=a+(n−1)⋅d.

Arithmetic progressions are widely used in real-life situations, such as in financial calculations, construction, and even in computer algorithms.

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5.2 Arithmetic Progressions: Key Concepts

An Arithmetic Progression (AP) has the following key properties:

• The first term of an AP is denoted by $a$a.
• The common difference $d$d is the difference between any two consecutive terms.
• The general form of an arithmetic progression is:

$$a, a + d, a + 2d, a + 3d, \dots$$a,a+d,a+2d,a+3d,…

Where:

• $a$a is the first term,
• $d$d is the common difference.

The $n$n-th term of an AP is given by the formula:$$a_n = a + (n – 1) \cdot d$$an​=a+(n−1)⋅d

Where:

• $a_n$an​ is the $n$n-th term,
• $n$n is the position of the term in the sequence.

Example:

If the first term $a = 2$a=2 and the common difference $d = 3$d=3, the first few terms of the AP are:$$2, 5, 8, 11, 14, \dots$$2,5,8,11,14,…

For this sequence:

• The 1st term ($a_1$a1​) is 2,
• The 2nd term ($a_2$a2​) is 5,
• The 3rd term ($a_3$a3​) is 8,
  and so on.

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5.3 nth Term of an AP

The $n$n-th term of an arithmetic progression can be found using the formula:$$a_n = a + (n – 1) \cdot d$$an​=a+(n−1)⋅d

Where:

• $a_n$an​ is the $n$n-th term,
• $a$a is the first term,
• $d$d is the common difference,
• $n$n is the position of the term in the progression.

Example:

Find the 10th term of the arithmetic progression $3, 6, 9, 12, 15, \dots$3,6,9,12,15,….

• Here, $a = 3$a=3 and $d = 3$d=3.
• Using the formula for the $n$n-th term:

$$a_{10} = 3 + (10 – 1) \cdot 3 = 3 + 27 = 30$$a10​=3+(10−1)⋅3=3+27=30

So, the 10th term of this AP is 30.

Another Example:

For the AP $5, 8, 11, 14, \dots$5,8,11,14,…, find the 7th term.

• $a = 5$a=5 and $d = 3$d=3.
• Using the formula:

$$a_7 = 5 + (7 – 1) \cdot 3 = 5 + 18 = 23$$a7​=5+(7−1)⋅3=5+18=23

So, the 7th term of this AP is 23.

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5.4 Sum of First n Terms of an AP

The sum of the first $n$n terms of an arithmetic progression is given by the formula:$$S_n = \frac{n}{2} \cdot \left[ 2a + (n – 1) \cdot d \right]$$Sn​=2n​⋅[2a+(n−1)⋅d]

Alternatively, the sum can also be expressed as:$$S_n = \frac{n}{2} \cdot (a + l)$$Sn​=2n​⋅(a+l)

Where:

• $S_n$Sn​ is the sum of the first $n$n terms,
• $a$a is the first term,
• $l$l is the last term of the progression,
• $d$d is the common difference.

Example:

Find the sum of the first 10 terms of the AP $2, 5, 8, 11, \dots$2,5,8,11,….

• Here, $a = 2$a=2, $d = 3$d=3, and $n = 10$n=10.

Using the formula for the sum of the first $n$n terms:$$S_{10} = \frac{10}{2} \cdot \left[ 2 \cdot 2 + (10 – 1) \cdot 3 \right]$$S10​=210​⋅[2⋅2+(10−1)⋅3] $$S_{10} = 5 \cdot \left[ 4 + 27 \right] = 5 \cdot 31 = 155$$S10​=5⋅[4+27]=5⋅31=155

So, the sum of the first 10 terms of the AP is 155.

Another Example:

Find the sum of the first 15 terms of the AP $7, 10, 13, 16, \dots$7,10,13,16,….

• $a = 7$a=7, $d = 3$d=3, and $n = 15$n=15.

First, find the 15th term using the $n$n-th term formula:$$a_{15} = 7 + (15 – 1) \cdot 3 = 7 + 42 = 49$$a15​=7+(15−1)⋅3=7+42=49

Now, using the sum formula:$$S_{15} = \frac{15}{2} \cdot (7 + 49) = \frac{15}{2} \cdot 56 = 15 \cdot 28 = 420$$S15​=215​⋅(7+49)=215​⋅56=15⋅28=420

So, the sum of the first 15 terms of the AP is 420.

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5.5 Summary

In this chapter, we have explored key concepts related to Arithmetic Progressions (AP):

• An Arithmetic Progression is a sequence of numbers with a constant difference between consecutive terms.
• The $n$n-th term of an AP can be calculated using the formula $a_n = a + (n – 1) \cdot d$an​=a+(n−1)⋅d, where $a$a is the first term, $d$d is the common difference, and $n$n is the position of the term.
• The sum of the first $n$n terms of an AP is given by the formula:

$$S_n = \frac{n}{2} \cdot \left[ 2a + (n – 1) \cdot d \right]$$Sn​=2n​⋅[2a+(n−1)⋅d]

• Arithmetic Progressions are widely used in real-life scenarios, including calculations related to finance, construction, and more.

MCQs Based on the “Arithmetic Progressions” Chapter:

1. What is the common difference in the AP: 5,9,13,17,…5, 9, 13, 17, \dots5,9,13,17,…?

a) 2
b) 3
c) 4
d) 5

Answer: b) 4

2. In the AP 3,7,11,15,…3, 7, 11, 15, \dots3,7,11,15,…, what is the 6th term?

a) 15
b) 17
c) 19
d) 21

Answer: c) 19

3. The sum of the first 10 terms of the AP 1,4,7,10,…1, 4, 7, 10, \dots1,4,7,10,… is:

a) 150
b) 160
c) 170
d) 180

Answer: b) 160

4. The sum of the first 20 terms of the AP 2,5,8,11,…2, 5, 8, 11, \dots2,5,8,11,… is:

a) 400
b) 420
c) 450
d) 500

**Answer