CBSE Notes Class 10 Maths Chapter 5 Arithmetic Progressions

CBSE Notes Class 10 Maths Chapter 5 – Arithmetic Progressions provide a clear and concise explanation of sequences in which each term increases or decreases by a constant value. In this chapter, students learn about the meaning of an Arithmetic Progression (AP), how to find the nth term, and how to calculate the sum of the first n terms using important formulas. These CBSE Notes help simplify concepts with step-by-step explanations, making revision quick and effective for board exam preparation

1.0Download CBSE Notes Class 10 Maths Chapter 5: Arithmetic Progressions - Free PDF!!

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2.0CBSE Maths Notes For Class 10 Chapter 5 Arithmetic Progressions - Revision Notes

Definition of Arithmetic Progression (AP)

The Arithmetic Progression is a pattern or sequence of things or numbers where the difference between each component of the sequence remains constant. This constant difference is termed a common difference (d).

Note: The common difference is the defining factor to determine whether a given sequence is an Arithmetic Progression or not. 

The General form of AP 

a, a+d, a+2d, a+3d ……..

d = a_n - a_n-1 

• a = First term. 
• d = Common difference. (The common difference can be negative, positive, or zero.)
• n = 1, 2, 3, 4, ………, n

General Form of n^th Term of Arithmetic Progression 

The formula for the nth term is 

a_n = a + (n - 1)d 

n = no. of terms in a given sequence. 

A_n = nth term or last term of AP. It is also denoted by l. 

Let’s understand AP with an example.

Example: Write the first four terms of an AP whose a = 2 and d = 3 are given below as follows. 

Solution: a₁ = 2 

a₂ = 2 + (2 - 1)3 = 2 + (1)3 = 2 + 3 = 5 

a₃ = 2 + (3 - 1)3 = 2 + (2)3 = 2 + 6 = 8

a₄ = 2 + (4 - 1)3 = 2 + (3)3 = 2 + 9 = 11

Example: What will be the 10^th term of multiple of 5? 

Solution: d = 10 - 5 = 5 

a₁₀ = 5 + (10-1)5 = 5 + (9)5 = 5 + 45 = 50 

Example: Find the last term of an AP 7, 14, 21,…... given that it contains 14 terms. 

Solution: d = 14 - 7 = 7

a₁₄ = 7 + (13-1)7 = 7 + (12)7 = 7 + 84 = 91.

Example: The sum of the 4th and 8th terms of an AP is 32, and the sum of the 6th and 10th terms is 52. Find the first term of the AP.

Solution: we have a₄ = 32 

a + 3d = 32 ……………………1

And a₈ = 52 

a + 7d = 52 ………………………….2

Eliminating equation 1 and 2 

a + 3d = 32

a + 7d = 52

4d = 20 

d = 5 

a + 3 5 = 32

a = 32 - 15 = 17

What if we sum up the terms of AP?

Let us once again consider the example that is given at the start where we get 500 interest every month. What will be the total amount collected after 6 months of investment? The total amount will be 8000. So, what we did here is simply sum up all the interest to the invested amount. 

The sum of n terms of AP is the same process where we sum up all the terms of AP. 

The formula for the Sum of nth terms of an AP 

${\mathrm{S}}_{\mathrm{n}}=\frac{n}{2}(2a+(n-1)d)$

Or alternatively:

${\mathrm{S}}_{\mathrm{n}}=\frac{n}{2}\left(a+a_n\right)$
Where:

• S_n is the sum of the first n terms
• a is the first term
• d is the common difference
• n is the number of terms
• a_n​ is the nth term

Points to note: 

• a₁ = s₁ 
• a_n = S_n - S_n-1 
• The sum of the first n positive integers is n(n-1)/2

Example: Find the sum of the first 20 multiples of 6. 

Solution: n = 20, a = 6, d = 6

$S_{20}=\frac{20}{2}(2\times 6+(20-1)6)=10(12+19\times 6)=1260$

3.0CBSE Maths Notes For Class 10 Chapter 5 Arithmetic Progressions - Key Notes

Arithmetic progressions

In general form, an arithmetic progression with first term 'a' and common difference 'd' can be represented as follows : a, a + d, a + 2d, a + 3d, a + 4d,.....

General term of an arithmetic progression

The nth term of an arithmetic progression is aₙ = a + (n − 1)d, where, a is the first term and d is the common difference of arithmetic progression.

rᵗʰ term of a finite arithmetic progression from the end

• If there are n terms in the arithmetic progression, then rᵗʰ term from the end = a + (n − r)d. So, rᵗʰ term from the end = a + (n − r)d
• Also, if ℓ is the last term of the arithmetic progression then rᵗʰ term from the end is the rᵗʰ term of an arithmetic progression whose first term is ℓ and common difference is −d. So, rᵗʰ term from the end = ℓ + (r − 1) (−d).

Sum of an arithmetic progression

Sum of n terms of an arithmetic progression

Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 [a + ℓ], where, a is the first term, d is the common difference and l is the last term of arithmetic progression.

Selection of terms in an AP

In general form, an arithmetic progression with first term 'a' and common difference 'd' can be represented as follows : a, a + d, a + 2d, a + 3d, a + 4d,.....

Points to remember

• If a constant is added or subtracted from each term of an AP, then the resulting sequence is also an AP with the same common difference.
• If each term of a given AP is multiplied or divided by a non-zero constant K, then the resulting sequence is also an AP with common difference Kd or d/K respectively, where d is the common difference of the given AP.
• In a finite AP, the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term.
• A sequence is an AP if it's nth term is a linear expression in n i.e., aₙ = An + B, where A, B are constants. In such a case, the coefficient of n is the common difference of the AP.
• If the terms of an AP are chosen at regular intervals, then they form an AP.
• The sum of first n odd natural numbers = n²
• The sum of first n natural numbers i.e. 1 + 2 + 3 + ... + n is usually written as Sₙ. $\sum n=\frac{n(n+1)}{2}$​
•  The sum of squares of first n natural numbers i.e. 1² + 2² + 3² + ... + n² is usually written as S₂.= $\sum n^2=\frac{n(n+1)(2n+1)}{6}$​
• The sum of cubes of first n natural numbers i.e. 1³ + 2³ + 3³ + ... + n³ is usually written as S₃.$\sum n^3={\left(\frac{n(n+1)}{2}\right)}^2=(S_n{)}^2$
• A sequence of non-zero numbers a₁, a₂, a₃,....,aₙ is said to be a geometric sequence or GP If $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=....$

4.0Key Features of CBSE Maths Notes for Class 10 of Chapter 5

• The notes are in line with the latest CBSE curriculum. 
• The language of the notes is easy to understand, making it ideal for self-learning. 
• Every concept of notes is provided with a solved problem to get a better understanding of AP. 
• The notes are made under the guidance of our experienced faculty to maintain accuracy.