**Arithmetic Progressions Introduction :**NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term, except for the first term. This fixed number is called the common difference (d) of the AP. The first term of an AP is denoted by ‘a’ and the nth term by ‘an’. The general formula for nth term of an AP is given as:

an = a + (n-1)d

Where, an = nth term of the AP a = first term of the AP d = common difference of the AP n = the number of terms in the AP

For example, consider the sequence 2, 5, 8, 11, 14, …. This is an arithmetic progression with first term (a) = 2 and common difference (d) = 3. To find the nth term of this AP, we can use the formula:

an = a + (n-1)d

So, the 10th term (n = 10) of this AP would be:

a10 = 2 + (10-1)3 = 2 + 27 = 29

APs are used in a variety of mathematical and real-life applications, such as in calculating the interest rate of a loan, in finding the distance covered by a moving object, in designing musical scales, and in many other fields.

## NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions

Arithmetic Progressions Class 10 has total of 4 exercises consists of 49 Problems. find the nth terms and the sum of n consecutive terms are important topics in this chapter 5.

**Topics and Sub Topics in NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions:**

Section Name |
Topic Name |

5 | Arithmetic Progressions |

5.1 | Introduction |

5.2 | Arithmetic Progressions |

5.3 | Nth Term Of An AP |

5.4 | Sum Of First N Terms Of An AP |

5.5 | Summary |

## Arithmetic Progressions formulas

Here are some of the important formulas related to Arithmetic Progressions (AP):

- nth term of an AP: The nth term (an) of an AP with first term (a) and common difference (d) can be found using the formula:

an = a + (n-1)d

- Sum of n terms of an AP: The sum (Sn) of the first n terms of an AP with first term (a) and common difference (d) can be found using the formula:

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

- Sum of first n natural numbers: The sum of the first n natural numbers is given by the formula:

1 + 2 + 3 + … + n = n(n+1)/2

- Sum of squares of first n natural numbers: The sum of the squares of the first n natural numbers is given by the formula:

1^2 + 2^2 + 3^2 + … + n^2 = n(n+1)(2n+1)/6

- Number of terms in an AP: The number of terms (n) in an AP with first term (a), last term (l) and common difference (d) can be found using the formula:

n = (l-a)/d + 1

These formulas can be used to solve problems related to APs and find the values of various parameters.

## What is Arithmetic Progressions

Arithmetic Progressions (AP) is a sequence of numbers in which each term is obtained by adding a fixed number (called the common difference) to the preceding term, except for the first term. In other words, an arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant value to the previous term.

For example, the sequence 2, 5, 8, 11, 14, … is an arithmetic progression with a first term of 2 and a common difference of 3. Each term in this sequence is obtained by adding 3 to the previous term.

APs are used in a variety of mathematical and real-life applications, such as in calculating the interest rate of a loan, in finding the distance covered by a moving object, in designing musical scales, and in many other fields. The study of APs is an important part of mathematics and is often included in school curricula.

## NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Ex 5.1

NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression Ex 5.1 is part of the NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression Exercise 5.1.

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 10 |

Subject |
Maths |

Chapter |
5 |

Chapter Name |
Arithmetic Progressions |

Exercise |
Ex 5.1 |

Number of Questions Solved |
4 |

Category |
NCERT Solutions |

## Use of Arithmetic Progressions

Arithmetic Progressions (AP) have a wide range of applications in mathematics and real-life situations. Some of the important uses of APs are:

- Financial calculations: APs are used to calculate the interest rates on loans, investments, and mortgages. For example, if the interest rate on a loan is an AP with a known common difference, then the total interest paid can be easily calculated.
- Distance and time calculations: APs are used to calculate the distance and time traveled by a moving object with uniform acceleration or deceleration. For example, the distance traveled by a car in a given time interval can be calculated using an AP formula.
- Music theory: APs are used in the construction of musical scales and chords. The frequencies of musical notes are arranged in an AP, with the common difference being a constant multiple of 2^(1/12).
- Statistics: APs are used in statistics to analyze and organize data. For example, the ages of a group of people can be arranged in an AP to study the distribution of ages.
- Mathematics: APs are used in various mathematical fields such as algebra, geometry, and calculus. They are used to model many real-world situations and to solve complex problems.

Thus, Arithmetic Progressions have a wide range of uses in various fields and are an important tool in mathematics and science.

## Methods of Arithmetic Progressions

There are several methods for working with Arithmetic Progressions (AP), including:

- Using the nth term formula: The nth term formula is a general formula used to find any term of an AP. It is given by:

an = a + (n-1)d

where a is the first term of the AP, d is the common difference, and n is the term number.

- Using the sum of n terms formula: The sum of n terms formula is used to find the sum of the first n terms of an AP. It is given by:

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

where a is the first term of the AP, d is the common difference, and n is the number of terms.

- Using the middle term formula: The middle term formula is used to find the middle term of an AP. It is given by:

am = a + (n/2-1)d

where a is the first term of the AP, d is the common difference, n is the number of terms, and am is the middle term.

- Using the common difference formula: The common difference formula is used to find the common difference of an AP when two terms and their positions are given. It is given by:

d = (an – am)/(n – m)

where a is the first term of the AP, d is the common difference, n is the position of the second term, m is the position of the first term, an is the nth term of the AP, and am is the mth term of the AP.

- Using the sum of two APs formula: The sum of two APs formula is used to find the sum of the terms of two APs with the same number of terms and common difference. It is given by:

S = n/2 [(a1 + b1) + (an + bn)]

where a1 and b1 are the first terms of the two APs, an and bn are the last terms of the two APs, n is the number of terms, and S is the sum of the terms.

These methods can be used to solve problems related to APs and find the values of various parameters.

## NCERT Solutions for Class 10 Maths Chapter 5 Ex 5.1 Arithmetic Progressions Questions

### Ex 5.1 Class 10 Maths Question 1.

(i) The list of numbers involved does not make an arithmetic progression. The fare increases by ₹ 8 after each additional km, but this increase is not a fixed constant. In an arithmetic progression, the difference between consecutive terms should be a constant, but here the difference between consecutive terms varies.

(ii) The list of numbers involved makes an arithmetic progression. When a vacuum pump removes 1/14 of the remaining air, it leaves 13/14 of the original air in the cylinder. So, the amount of air remaining after each cycle is obtained by multiplying the previous amount of air by 13/14. Thus, the list of numbers involved forms a geometric progression, and its logarithm will form an arithmetic progression.

(iii) The list of numbers involved makes an arithmetic progression. The cost of digging the well increases by ₹ 50 for each subsequent meter, which is a fixed constant. So, the difference between consecutive terms is constant, and hence, the list of numbers forms an arithmetic progression.

(iv) The list of numbers involved does not make an arithmetic progression. The amount of money in the account after each year is not obtained by adding a fixed constant to the previous amount. It is obtained by compounding the interest on the previous amount, which leads to an exponential growth rather than a linear one. So, the list of numbers involved forms a geometric progression.

### Ex 5.1 Class 10 Maths Question 2.

(i) a = 10, d = 10

(ii) a = -2, d = 0

(iii) a = 4, d = -3

(iv) a = -1, d = 12

(v) a = -1.25, d = -0.25

We can use the formula for nth term of an arithmetic progression to find the first four terms of the given APs:

(i) a = 10, d = 10 The nth term formula is: an = a + (n – 1)d Using this formula, we get:

- first term, a1 = 10
- second term, a2 = 10 + (2 – 1)10 = 20
- third term, a3 = 10 + (3 – 1)10 = 30
- fourth term, a4 = 10 + (4 – 1)10 = 40

So, the first four terms of this AP are 10, 20, 30, 40.

(ii) a = -2, d = 0 The nth term formula is: an = a + (n – 1)d Using this formula, we get:

- first term, a1 = -2
- second term, a2 = -2 + (2 – 1)0 = -2
- third term, a3 = -2 + (3 – 1)0 = -2
- fourth term, a4 = -2 + (4 – 1)0 = -2

So, the first four terms of this AP are -2, -2, -2, -2.

(iii) a = 4, d = -3 The nth term formula is: an = a + (n – 1)d Using this formula, we get:

- first term, a1 = 4
- second term, a2 = 4 + (2 – 1)(-3) = 1
- third term, a3 = 4 + (3 – 1)(-3) = -2
- fourth term, a4 = 4 + (4 – 1)(-3) = -5

So, the first four terms of this AP are 4, 1, -2, -5.

(iv) a = -1, d = 12 The nth term formula is: an = a + (n – 1)d Using this formula, we get:

- first term, a1 = -1
- second term, a2 = -1 + (2 – 1)12 = 11
- third term, a3 = -1 + (3 – 1)12 = 23
- fourth term, a4 = -1 + (4 – 1)12 = 35

So, the first four terms of this AP are -1, 11, 23, 35.

(v) a = -1.25, d = -0.25 The nth term formula is: an = a + (n – 1)d Using this formula, we get:

- first term, a1 = -1.25
- second term, a2 = -1.25 + (2 – 1)(-0.25) = -1.5
- third term, a3 = -1.25 + (3 – 1)(-0.25) = -1.75
- fourth term, a4 = -1.25 + (4 – 1)(-0.25) = -2

So, the first four terms of this AP are -1.25, -1.5, -1.75, -2.

### Ex 5.1 Class 10 Maths Question 3.

For the following APs, write the first term and the common difference:

(i) 3, 1, -1, -3, ……

(ii) -5, -1, 3, 7, ……

(iii) 13 , 53 , 93, 133 , ……..

(iv) 0.6, 1.7, 2.8, 3.9, …….

We can observe the given sequence of numbers to find the common difference and the first term of the corresponding arithmetic progressions (APs).

(i) 3, 1, -1, -3, …… The common difference is d = -2 (subtract 2 from each term to get the next term). The first term is a1 = 3.

(ii) -5, -1, 3, 7, …… The common difference is d = 4 (add 4 to each term to get the next term). The first term is a1 = -5.

(iii) 13 , 53 , 93, 133 , …….. The common difference is d = 40 (add 40 to each term to get the next term). The first term is a1 = 13.

(iv) 0.6, 1.7, 2.8, 3.9, ……. The common difference is d = 1.1 (add 1.1 to each term to get the next term). The first term is a1 = 0.6.

### Ex 5.1 Class 10 Maths Question 4.

(i) 2, 4, 8, 16, …….

(ii) 2, 52 , 3, 72 , …….

(iii) -1.2, -3.2, -5.2, -7.2, ……

(iv) -10, -6, -2,2, …..

(v) 3, 3 + 2–√, 3 + 22–√, 3 + 32–√, …..

(vi) 0.2, 0.22, 0.222, 0.2222, ……

(vii) 0, -4, -8, -12, …..

(viii) −12 , −12 , −12 , −12 , …….

(ix) 1, 3, 9, 27, …….

(x) a, 2a, 3a, 4a, …….

(xi) a, a2, a3, a4, …….

(xii) 2–√, 8–√, 18−−√, 32−−√, …..

(xiii) 3–√, 6–√, 9–√, 12−−√, …..

(xiv) 12, 32, 52, 72, ……

(xv) 12, 52, 72, 73, ……

(i) 2, 4, 8, 16, ……. This is an AP with a common difference of d = 4. Three more terms are 32, 64, 128.

(ii) 2, 52 , 3, 72 , ……. This is not an AP as the common difference is not constant.

(iii) -1.2, -3.2, -5.2, -7.2, …… This is an AP with a common difference of d = -2. Three more terms are -9.2, -11.2, -13.2.

(iv) -10, -6, -2, 2, ….. This is an AP with a common difference of d = 4. Three more terms are 6, 10, 14.

(v) 3, 3 + 2–√, 3 + 22–√, 3 + 32–√, ….. This is not an AP as the common difference is not constant.

(vi) 0.2, 0.22, 0.222, 0.2222, …… This is not an AP as the common difference is not constant.

(vii) 0, -4, -8, -12, ….. This is an AP with a common difference of d = -4. Three more terms are -16, -20, -24.

(viii) −12 , −12 , −12 , −12 , ……. This is an AP with a common difference of d = 0. Three more terms are -12, -12, -12.

(ix) 1, 3, 9, 27, ……. This is not an AP as the common difference is not constant.

(x) a, 2a, 3a, 4a, ……. This is an AP with a common difference of d = a. Three more terms are 5a, 6a, 7a.

(xi) a, a2, a3, a4, ……. This is not an AP as the common difference is not constant.

(xii) 2–√, 8–√, 18−−√, 32−−√, ….. This is not an AP as the common difference is not constant.

(xiii) 3–√, 6–√, 9–√, 12−−√, ….. This is an AP with a common difference of d = 3√. Three more terms are 15−√, 18−√, 21−√.

(xiv) 12, 32, 52, 72, …… This is an AP with a common difference of d = 20. Three more terms are 92, 112, 132.

(xv) 12, 52, 72, 73, …… This is not an AP as the common difference is not constant.

## Arithmetic Progression (AP)

Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is a constant, known as the common difference. This constant difference is denoted by ‘d’. The first term of the AP is denoted by ‘a’, and the nth term of the AP is denoted by ‘an’.

The general formula to find the nth term of an AP is given by:

an = a + (n-1)d

Where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the nth term.

The sum of the first ‘n’ terms of an AP can be found using the following formula:

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

Where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the number of terms.

APs are used in many mathematical and real-world applications, such as in calculating interest rates, in determining the position and velocity of moving objects, and in solving various problems related to arithmetic and algebraic operations.

### Arithmetic Progression (AP) some examples

Here are some examples of Arithmetic Progressions (APs):

Example 1: 2, 4, 6, 8, 10, …

In this AP, the first term is ‘2’, and the common difference is ‘2’, as we add 2 to each term to get the next one.

Example 2: -3, -1, 1, 3, 5, …

In this AP, the first term is ‘-3’, and the common difference is ‘2’, as we add 2 to each term to get the next one.

Example 3: 5.5, 8, 10.5, 13, 15.5, …

In this AP, the first term is ‘5.5’, and the common difference is ‘2.5’, as we add 2.5 to each term to get the next one.

Example 4: -10, -5, 0, 5, 10, …

In this AP, the first term is ‘-10’, and the common difference is ‘5’, as we add 5 to each term to get the next one.

Example 5: 1/2, 1, 3/2, 2, 5/2, …

In this AP, the first term is ‘1/2’, and the common difference is ‘1/2’, as we add 1/2 to each term to get the next one.

These are just a few examples of Arithmetic Progressions. There are many more APs in mathematics and the real world.

### Formula for common Difference (d)

The formula for the common difference (d) in an arithmetic progression (AP) is:

d = (a_n – a_1) / (n – 1)

where d is the common difference, a_n is the nth term of the AP, a_1 is the first term of the AP, and n is the total number of terms in the AP.

### nth Term (or General Term) of an Arithmetic Progressions

The nth term (or general term) of an arithmetic progression (AP) can be found using the following formula:

a_n = a_1 + (n-1)d

where a_n is the nth term of the AP, a_1 is the first term of the AP, n is the number of terms, and d is the common difference.

This formula can be derived by noticing that in an AP, the difference between consecutive terms is constant. Therefore, to find the nth term, we can add the common difference (d) to the (n-1)th term. Since the first term is a_1, we can express the nth term as a_1 + (n-1)d.

### Sum of the FIRST ‘n’ Terms of an A.P.

The sum of the first n terms of an arithmetic progression (AP) can be found using the following formula:

S_n = (n/2)(a_1 + a_n)

where S_n is the sum of the first n terms, a_1 is the first term of the AP, a_n is the nth term of the AP, and n is the number of terms in the AP.

This formula can be derived by using the formula for the nth term of an AP, which is a_n = a_1 + (n-1)d. By substituting this expression for a_n in the sum of the first n terms, we get:

S_n = a_1 + (a_1 + d) + (a_1 + 2d) + … + (a_1 + (n-1)d)

S_n = n(a_1) + d(1 + 2 + … + (n-1))

The sum of the first (n-1) positive integers can be found using the formula n(n-1)/2. Therefore, we get:

S_n = n(a_1) + d(n(n-1)/2)

Simplifying this expression, we get:

S_n = (n/2)(2a_1 + (n-1)d)

which is the same as the formula given above.

### Arithmetic Mean Between Two Numbers

The arithmetic mean between two numbers is the number that is exactly midway between them.

Let the two numbers be a and b, then the arithmetic mean between them is given by:

Arithmetic Mean = (a + b)/2

For example, the arithmetic mean between 3 and 7 is:

Arithmetic Mean = (3 + 7)/2 = 5

So, the arithmetic mean between 3 and 7 is 5.

## FAQs About NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression (AP)

- What is an arithmetic progression (AP)? An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.
- What is the formula for the nth term of an arithmetic progression? The formula for the nth term of an arithmetic progression is: an = a1 + (n-1)d where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
- What is the formula for the sum of the first n terms of an arithmetic progression? The formula for the sum of the first n terms of an arithmetic progression is: Sn = (n/2)(a1 + an) where Sn is the sum of the first n terms, a1 is the first term, an is the nth term, and n is the number of terms.
- What is the formula for the common difference of an arithmetic progression? The formula for the common difference of an arithmetic progression is: d = (an – a1)/(n-1) where d is the common difference, an is the nth term, a1 is the first term, and n is the number of terms.
- How do you find the arithmetic mean between two numbers? The arithmetic mean between two numbers a and b is given by: Arithmetic Mean = (a + b)/2
- How do you identify an arithmetic progression? To identify an arithmetic progression, you need to check if the difference between any two consecutive terms is constant. If the difference is the same for all pairs of consecutive terms, then the sequence is an arithmetic progression.
- What is the difference between an arithmetic progression and a geometric progression? An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. A geometric progression, on the other hand, is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor

Q: Are the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions free to download?

A: Yes, the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions are free to download. They are available in PDF format on various educational websites and can be downloaded for free.

Q: Are the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions accurate?

A: Yes, the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions are accurate. They have been prepared by subject matter experts and are based on the latest CBSE syllabus.

Q: Are the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions helpful for board exams?

A: Yes, the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions are helpful for board exams. They provide a comprehensive understanding of the chapter and help students to solve problems related to Arithmetic Progressions with ease.

Q: Are the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions useful for competitive exams?

A: Yes, the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions are useful for competitive exams. They cover all the important topics related to Arithmetic Progressions and provide a strong foundation for solving problems related to AP in various competitive exams like JEE, NEET, etc.

Q: Can I rely on NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions as my only source of preparation?

A: While the NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions provide a comprehensive understanding of the chapter, it is always recommended to refer to other reference books and study materials for additional practice and better preparation

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