# Class 10 Arithmetic Progression

The concept of progression series is introduced in class 10 Maths Arithmetic Progression (AP) chapter. Being one of the most essential chapters of class 10 maths syllabus, it is important to be well-versed with the intricacies of this one to excel in the board examination. Once, you are through with the concept of AP, you can easily solve the complex questions that are a part of competitive exams as well. In this blog, we will discover more about arithmetic progression, its formula, solved examples, etc. We shall also understand how to find the nth term and the sum of the consecutive terms. So, let’s begin!

## Sequences, Series and Progressions

• A sequence is a finite or infinite set of integers that follow a pattern. For example, the series is infinite: 1, 2, 3, 4, 5… Natural number sequence.
• The total of the components in the matching sequence is referred to as a series. For example, 1+2+3+4+5…. is a natural number sequence. Each number in a sequence or series is referred to as a term.
• A progression is a sequence in which the general term can be can be expressed using a mathematical formula.

## Arithmetic Progression

Let us begin by understanding what arithmetic progressions is, Arithmetic Progression can be simply defined as a series of numbers wherein the difference between any of the 2 consecutive numbers is always the same. For Example, In a given series of natural numbers, i.e., 6, 7,8,9,10, …n, we can say that the series is an arithmetic progression as the difference between 2 consecutives terms (7 and 8 or 6 and 7) is equal to 1 (7-6).

If we consider a series of odd numbers and one of even numbers, then it will be an AP as the common difference between any two consecutive terms of the respective series is equal to 2.

According to the class 10 arithmetic progression chapter, In an arithmetic progression, the series refers to the sum of elements of the successive numbers. The difference between one term and its next term is always constant in AP.

We can say, if we just add similar value or numbers to the previous numbers every time, the result will be constant. This property can be understood from the below-mentioned formula:

a1 + an = a2 + an-1 = … = ak + an-k+1 and an = ½(an-1 + an+1)

Now that we are through the theoretical understanding of class 10 arithmetic progression, let us move forward and understand the concept through solved examples.

Example 1: Find the AP given by x + b, x + 3b, x + 5b,…
Solution 1:  Here a = x + b ,
d = x + 3b – (x + b)
d = x + 3b – x – b
∴ d = 2b

The general term of an AP is given by the formula
an = a + (n – 1) d
an = x + b + (n – 1)2b
∴ an = x + (2n – 1)b

## Arithmetic Progression and the Sequence of Numbers

In our class 10 arithmetic progression notes, the next topic is the sequence of numbers in AP. When the difference between two successive numbers is always constant, then it can be termed as a sequence. For Example:

• 1, 2, 3, 4, 5… is a sequence, which is arithmetic progression and has a common difference of 1
• 3, 5, 7, 9, 11… is another common sequence that is an arithmetic progression and has a common difference of 2
• In a given sequence, 10, 20, 30, 40, 50… that is arithmetic progression has a  common difference of 10

The arithmetic sequence can also be written as:
{ a, a + b, a + 2b, a + 3b, a + 4b, ….}
Where,
a is the first term, and
b is the difference between the terms

Example 2:  Which term of the A.P. 3,8,13 …is 78?
Solution: Here an = a + (n – 1) d = 78
a= 3, d = 8- 3 = 5
Therefore,
3 + (n -1) (5) = 78
(n-1) * 5 = 78 – 3 = 75
n – 1 = 75/5 = 15
n = 15 + 1 = 1
Hence, 78 is the 16th term.

Example 3: Which term of the A.P. 3, 15, 27, 39 … will be 132 more than its 54th term?
Solution: Given series is 3, 15, 27, 39…
Here, a =3, d= 15-3 = 12
Since an = ak = (n-k) d
an – a54 = (n-54) 12
132 = 12n – 54 * 12    …..(since an – a 54 = 132 given)
12 n = 132 + 54 * 12 = 12 (11+ 54)
n= 11 + 54 = 65

Commerce Subjects in Class 11

## Progression and Its Types

As we move further with the class 10 arithmetic progression, we will now learn the various types of progression. Progression is a kind of special sequence of numbers. This makes it possible to determine or obtain the formula of the nth term. Majorly, progressions can be bifurcated into three categories such as:

• Harmonic Progression (HP)
• Geometric Progression (GP)
• Arithmetic Progressions (AP)

The arithmetic progressions (AP) is basically the simplest progression sequence used. It is a difference noticed between any 2 successive terms that are always constant in AP. (towncville.com)

Here are some Mental Maths: Tips & Tricks to improve your calculations!

## Notation in Arithmetic Progressions

Through our class 10 arithmetic progression notes, you are familiar with the definition of AP and when a fixed number can be added to any of its terms which are also known as the common difference in AP. Three major terms denoted in arithmetic progressions are:

• Common difference
• nth term
• Sum of the nth term

All of them represent the properties of an arithmetic progression.

Career Options in Commerce Without Maths

## Uses of Arithmetic Progressions

Arithmetic progressions are used to set or generalize a pattern in observable studies.

To calculate the sum of an AP, it is vital to have the first term, common differences between the terms and the number of all the terms. The general formula to find the nth term is:

an = a + (n-1) × d

Alternative formula is, S = n/2 [ 2a + (n-1) × d]

## The nth Term of the Arithmetic Progression

The formula for obtaining the nth term of an AP is a = a + (n+1) d.

In this formula, a is the initial term in the arithmetic sequence while d is the common difference between the terms.

Both of these values can either be negative or a positive integer. It is because the actual value of the arithmetic sequence can be a negative value, and the difference between any two sequences can be a negative integer.

n is also a number in terms of an arithmetic sequence. As negative integers are not used to count anything, the value of the nth term in the formula will not be a negative integer.

## Sum of the First ‘n’ Terms of an AP

In the formula of AP the sum of the progression term with ‘n’ term, ‘a’ as the first term, and ‘d’ as the common difference are given as:

S = n/2 [2a + (n-1) d]

If l is the nth term of a finite A.P., then The sum of all terms of the AP is given by:

S = n(a + l) / 2

It is worth noting that the sum of the first n positive integers is given by:

S = [n(n + 1)] / 2

If a, b, c are in AP. Then b is called the arithmetic mean of a and c and is given by

b = (a + c) / 2

Example 4: Find the 15th term of the arithmetic progression 3, 9, 15, 21,….?
Solution: In the given AP, a = 3, d = (9 – 3) = 6, n =15
T15 = a + (n – 1)d
= 3 + (15 -1)6
= 3 + 84 = 87

Take This Maths Quiz If You Consider Yourself Genius!

## Class 10 Arithmetic Progression NCERT PDF

You can download the NCERT PDF for class 10 Arithmetic Progression from Here!

## Class 10 Arithmetic Progression Notes PDF

To help you with the chapter, we have attached below a pdf of it-

## Class 10 Arithmetic Progression Questions with Solution

In a certain AP,  the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Given, a 24 = 2210
224 = a + 23d and a10 = a + 9d
a72 = a + 710
a34 = a + 33d
a24 = 2010—————–(Given)
a + 23d = 2(a +9d)
a + 23d = 2a + 18d
a – 5d = 0
a = 5d
a72 = 2 a34
a + 710 = 2(a + 33d)a + 710 = 2a + 66d
a – 5d = 0
a = 5d
(ii)from, (1) and (2) 272 = 2a34

If the gth term of an A.P. is zero, prove that its 29th term is double the 19th term.

Given: gth term = 0
a, + 8d = 0
a29 = a1 + 28d = a +8d + 200 = 0 + 20d = 20d
219 = a1 + 18d = a, + 80 + 100 = 0 + 10d = 10d
a29 = 2a19

Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . .

For the given, A.P. 11, 8, 5, 2, …
First-term, a = 11
Common difference, d = a2 − a1 = 8 − 11 = −3
Let −150 be the nth term of this A.P.
As we know, for an A.P.,
an = a + (n − 1) d
-150 = 11 + (n – 1)(-3)
-150 = 11 – 3n + 3
-164 = -3n
n = 164/3
Clearly, n is not an integer but a fraction.
Therefore, – 150 is not a term of this A.P.

How many terms of the AP: 24, 21, 18, . . . must be taken so that their sum is 78

Here, a = 24, d = 21 – 24 = –3, Sn = 78. We need to find n.
We know that;
Sn = n/2[2a + (n-1)d]So, 78 = n/2 [48 + (n-1)(-3)]= n/2 [51 – 3n]3n2 – 51n +156 = 0
n2 – 17n + 52 = 0
(n-4) (n-13) = 0
n = 4 or 13
Both values of n are admissible. So, the number of terms is either 4 or 13.

Which term of the AP: 3, 8, 13, 18, …, is 78?

Given: 3, 8, 13, 18, …………

a = 3, d = 8 -3 = 5

Let nth term be 78

a= 78

a + (n – 1)d = 78

3 + (n – 1)5 = 78

(n – 1)5 = 78 – 3 = 75

n – 1 = 15

n = 15 + 1 = 16

Hence a16  = 78

Find the number of terms in the AP: 7, 13, 19, …, 205

Here, a = 7, d = 13 – 7 = 6 and l = 205

Using the formula, l = a + (n – 1)d,

We get:

205 = 7 + (n – 1) x 6

(n – 1) = (205 – 7) / 6

n – 1 = 33

n = 33 + 1 = 34

Hence, the number of terms in the given AP is 34

## Extra Questions

For a better understanding of the class 10 Arithmetic Progression, you may solve the following extra questions-

• Two A. P.’s has the same common difference. The difference between their 100th terms is 111222333. What is the difference between their Millionth terms?
• The 10th and 18th terms of an A. P. are 41 and 73 respectively. Find 26th term.
• If an A. P. consists of n terms with first term a and nth term l show that the sum of the nth term from the beginning and the mth term from the end is (a + l).
• Find the second term and nth term of an A. P. whose 6th term is 12 and the 8th term is 22.
• If the nth term of the A. P. 9, 7, 5…. is same as the nth term of the A. P. 15, 12, 9…… find n.
• The nth term of an A. P. is 6n + 11. Find the common difference.
• The general term of a sequence is given by an = -4n + 15. Is the sequence an A. P.? If so, find its 15th term and the common difference.
• The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
• Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
• If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly find the 3rd, the 10th and the nth terms.

## Class 10 Arithmetic Progression HOTS Questions

1.  If  (an+1  + bn+1) / (an + bn) is the arithmetic mean between ‘a’ and ‘b’, then, find the value of ‘n’.
2.  If 1 / b + c , 1 / c + a , 1 / a + b are in A.P., prove that a2, b2, c2 are also in A.P.
3. Find three numbers in A.P. whose sum is 21 and their product is 231.
4. Find p and q such that: 2p, 2p, q, p + 4q, 35 are in AP
5. For what value of p, are (2p – 1), 7 and (11/2)p three consecutive terms of an AP?

## Class 10 Arithmetic Proression Worksheet

For your practice, here is a worksheet-

## Class 10 Arithmetic Progression PPT

Thus, we hope that through this blog about class 10 Maths arithmetic progression, you are now through with one of the most essential topics of class 10th. Reach out to our Leverage Edu experts for assistance regarding career guidance and stream selection.

The concept of progression series is introduced in class 10 Maths Arithmetic Progression (AP) chapter. Being one of the most essential chapters of class 10 maths syllabus, it is important to be well-versed with the intricacies of this one to excel in the board examination. Once, you are through with the concept of AP, you can easily solve the complex questions that are a part of competitive exams as well. In this blog, we will discover more about arithmetic progression, its formula, solved examples, etc. We shall also understand how to find the nth term and the sum of the consecutive terms. So, let’s begin!

## Sequences, Series and Progressions

• A sequence is a finite or infinite set of integers that follow a pattern. For example, the series is infinite: 1, 2, 3, 4, 5… Natural number sequence.
• The total of the components in the matching sequence is referred to as a series. For example, 1+2+3+4+5…. is a natural number sequence. Each number in a sequence or series is referred to as a term.
• A progression is a sequence in which the general term can be can be expressed using a mathematical formula.

## Arithmetic Progression

Let us begin by understanding what arithmetic progressions is, Arithmetic Progression can be simply defined as a series of numbers wherein the difference between any of the 2 consecutive numbers is always the same. For Example, In a given series of natural numbers, i.e., 6, 7,8,9,10, …n, we can say that the series is an arithmetic progression as the difference between 2 consecutives terms (7 and 8 or 6 and 7) is equal to 1 (7-6).

If we consider a series of odd numbers and one of even numbers, then it will be an AP as the common difference between any two consecutive terms of the respective series is equal to 2.

According to the class 10 arithmetic progression chapter, In an arithmetic progression, the series refers to the sum of elements of the successive numbers. The difference between one term and its next term is always constant in AP.

We can say, if we just add similar value or numbers to the previous numbers every time, the result will be constant. This property can be understood from the below-mentioned formula:

a1 + an = a2 + an-1 = … = ak + an-k+1 and an = ½(an-1 + an+1)

Now that we are through the theoretical understanding of class 10 arithmetic progression, let us move forward and understand the concept through solved examples.

Example 1: Find the AP given by x + b, x + 3b, x + 5b,…
Solution 1:  Here a = x + b ,
d = x + 3b – (x + b)
d = x + 3b – x – b
∴ d = 2b

The general term of an AP is given by the formula
an = a + (n – 1) d
an = x + b + (n – 1)2b
∴ an = x + (2n – 1)b

## Arithmetic Progression and the Sequence of Numbers

In our class 10 arithmetic progression notes, the next topic is the sequence of numbers in AP. When the difference between two successive numbers is always constant, then it can be termed as a sequence. For Example:

• 1, 2, 3, 4, 5… is a sequence, which is arithmetic progression and has a common difference of 1
• 3, 5, 7, 9, 11… is another common sequence that is an arithmetic progression and has a common difference of 2
• In a given sequence, 10, 20, 30, 40, 50… that is arithmetic progression has a  common difference of 10

The arithmetic sequence can also be written as:
{ a, a + b, a + 2b, a + 3b, a + 4b, ….}
Where,
a is the first term, and
b is the difference between the terms

Example 2:  Which term of the A.P. 3,8,13 …is 78?
Solution: Here an = a + (n – 1) d = 78
a= 3, d = 8- 3 = 5
Therefore,
3 + (n -1) (5) = 78
(n-1) * 5 = 78 – 3 = 75
n – 1 = 75/5 = 15
n = 15 + 1 = 1
Hence, 78 is the 16th term.

Example 3: Which term of the A.P. 3, 15, 27, 39 … will be 132 more than its 54th term?
Solution: Given series is 3, 15, 27, 39…
Here, a =3, d= 15-3 = 12
Since an = ak = (n-k) d
an – a54 = (n-54) 12
132 = 12n – 54 * 12    …..(since an – a 54 = 132 given)
12 n = 132 + 54 * 12 = 12 (11+ 54)
n= 11 + 54 = 65

Commerce Subjects in Class 11

## Progression and Its Types

As we move further with the class 10 arithmetic progression, we will now learn the various types of progression. Progression is a kind of special sequence of numbers. This makes it possible to determine or obtain the formula of the nth term. Majorly, progressions can be bifurcated into three categories such as:

• Harmonic Progression (HP)
• Geometric Progression (GP)
• Arithmetic Progressions (AP)

The arithmetic progressions (AP) is basically the simplest progression sequence used. It is a difference noticed between any 2 successive terms that are always constant in AP. (towncville.com)

Here are some Mental Maths: Tips & Tricks to improve your calculations!

## Notation in Arithmetic Progressions

Through our class 10 arithmetic progression notes, you are familiar with the definition of AP and when a fixed number can be added to any of its terms which are also known as the common difference in AP. Three major terms denoted in arithmetic progressions are:

• Common difference
• nth term
• Sum of the nth term

All of them represent the properties of an arithmetic progression.

Career Options in Commerce Without Maths

## Uses of Arithmetic Progressions

Arithmetic progressions are used to set or generalize a pattern in observable studies.

To calculate the sum of an AP, it is vital to have the first term, common differences between the terms and the number of all the terms. The general formula to find the nth term is:

an = a + (n-1) × d

Alternative formula is, S = n/2 [ 2a + (n-1) × d]

## The nth Term of the Arithmetic Progression

The formula for obtaining the nth term of an AP is a = a + (n+1) d.

In this formula, a is the initial term in the arithmetic sequence while d is the common difference between the terms.

Both of these values can either be negative or a positive integer. It is because the actual value of the arithmetic sequence can be a negative value, and the difference between any two sequences can be a negative integer.

n is also a number in terms of an arithmetic sequence. As negative integers are not used to count anything, the value of the nth term in the formula will not be a negative integer.

## Sum of the First ‘n’ Terms of an AP

In the formula of AP the sum of the progression term with ‘n’ term, ‘a’ as the first term, and ‘d’ as the common difference are given as:

S = n/2 [2a + (n-1) d]

If l is the nth term of a finite A.P., then The sum of all terms of the AP is given by:

S = n(a + l) / 2

It is worth noting that the sum of the first n positive integers is given by:

S = [n(n + 1)] / 2

If a, b, c are in AP. Then b is called the arithmetic mean of a and c and is given by

b = (a + c) / 2

Example 4: Find the 15th term of the arithmetic progression 3, 9, 15, 21,….?
Solution: In the given AP, a = 3, d = (9 – 3) = 6, n =15
T15 = a + (n – 1)d
= 3 + (15 -1)6
= 3 + 84 = 87

Take This Maths Quiz If You Consider Yourself Genius!

## Class 10 Arithmetic Progression NCERT PDF

You can download the NCERT PDF for class 10 Arithmetic Progression from Here!

## Class 10 Arithmetic Progression Notes PDF

To help you with the chapter, we have attached below a pdf of it-

## Class 10 Arithmetic Progression Questions with Solution

In a certain AP,  the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Given, a 24 = 2210
224 = a + 23d and a10 = a + 9d
a72 = a + 710
a34 = a + 33d
a24 = 2010—————–(Given)
a + 23d = 2(a +9d)
a + 23d = 2a + 18d
a – 5d = 0
a = 5d
a72 = 2 a34
a + 710 = 2(a + 33d)a + 710 = 2a + 66d
a – 5d = 0
a = 5d
(ii)from, (1) and (2) 272 = 2a34

If the gth term of an A.P. is zero, prove that its 29th term is double the 19th term.

Given: gth term = 0
a, + 8d = 0
a29 = a1 + 28d = a +8d + 200 = 0 + 20d = 20d
219 = a1 + 18d = a, + 80 + 100 = 0 + 10d = 10d
a29 = 2a19

Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . .

For the given, A.P. 11, 8, 5, 2, …
First-term, a = 11
Common difference, d = a2 − a1 = 8 − 11 = −3
Let −150 be the nth term of this A.P.
As we know, for an A.P.,
an = a + (n − 1) d
-150 = 11 + (n – 1)(-3)
-150 = 11 – 3n + 3
-164 = -3n
n = 164/3
Clearly, n is not an integer but a fraction.
Therefore, – 150 is not a term of this A.P.

How many terms of the AP: 24, 21, 18, . . . must be taken so that their sum is 78

Here, a = 24, d = 21 – 24 = –3, Sn = 78. We need to find n.
We know that;
Sn = n/2[2a + (n-1)d]So, 78 = n/2 [48 + (n-1)(-3)]= n/2 [51 – 3n]3n2 – 51n +156 = 0
n2 – 17n + 52 = 0
(n-4) (n-13) = 0
n = 4 or 13
Both values of n are admissible. So, the number of terms is either 4 or 13.

Which term of the AP: 3, 8, 13, 18, …, is 78?

Given: 3, 8, 13, 18, …………

a = 3, d = 8 -3 = 5

Let nth term be 78

a= 78

a + (n – 1)d = 78

3 + (n – 1)5 = 78

(n – 1)5 = 78 – 3 = 75

n – 1 = 15

n = 15 + 1 = 16

Hence a16  = 78

Find the number of terms in the AP: 7, 13, 19, …, 205

Here, a = 7, d = 13 – 7 = 6 and l = 205

Using the formula, l = a + (n – 1)d,

We get:

205 = 7 + (n – 1) x 6

(n – 1) = (205 – 7) / 6

n – 1 = 33

n = 33 + 1 = 34

Hence, the number of terms in the given AP is 34

## Extra Questions

For a better understanding of the class 10 Arithmetic Progression, you may solve the following extra questions-

• Two A. P.’s has the same common difference. The difference between their 100th terms is 111222333. What is the difference between their Millionth terms?
• The 10th and 18th terms of an A. P. are 41 and 73 respectively. Find 26th term.
• If an A. P. consists of n terms with first term a and nth term l show that the sum of the nth term from the beginning and the mth term from the end is (a + l).
• Find the second term and nth term of an A. P. whose 6th term is 12 and the 8th term is 22.
• If the nth term of the A. P. 9, 7, 5…. is same as the nth term of the A. P. 15, 12, 9…… find n.
• The nth term of an A. P. is 6n + 11. Find the common difference.
• The general term of a sequence is given by an = -4n + 15. Is the sequence an A. P.? If so, find its 15th term and the common difference.
• The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
• Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
• If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly find the 3rd, the 10th and the nth terms.

## Class 10 Arithmetic Progression HOTS Questions

1.  If  (an+1  + bn+1) / (an + bn) is the arithmetic mean between ‘a’ and ‘b’, then, find the value of ‘n’.
2.  If 1 / b + c , 1 / c + a , 1 / a + b are in A.P., prove that a2, b2, c2 are also in A.P.
3. Find three numbers in A.P. whose sum is 21 and their product is 231.
4. Find p and q such that: 2p, 2p, q, p + 4q, 35 are in AP
5. For what value of p, are (2p – 1), 7 and (11/2)p three consecutive terms of an AP?

## Class 10 Arithmetic Proression Worksheet

For your practice, here is a worksheet-

## Class 10 Arithmetic Progression PPT

Thus, we hope that through this blog about class 10 Maths arithmetic progression, you are now through with one of the most essential topics of class 10th. Reach out to our Leverage Edu experts for assistance regarding career guidance and stream selection.