NCERT Solutions for Class 10 Maths Chapter 5 Exercise 5.4 - Arithmetic Progressions

NCERT Solutions for Class 10 Maths Chapter 5 Exercise 5.4 - Arithmetic Progressions

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Komal MiglaniUpdated on 02 Jun 2025, 02:34 PM IST

A progression is a sequence (or a set) of numbers that follow a specific pattern of repetition. We can define Arithmetic Progression as a set of numbers where the difference between any two consecutive terms remains the same throughout the series. This difference between the two terms is called the common difference of the AP. The general form of an AP is a, a + d, a + 2d, .... , where d is the common difference, and a is the first term.

This Story also Contains

  1. Arithmetic Progressions Class 10 Chapter 5 Exercise: 5.4
  2. Topics Covered in Chapter 5, Arithmetic Progression: Exercise 5.4
  3. NCERT Solutions of Class 10 Subject Wise
  4. NCERT Exemplar Solutions of Class 10 Subject Wise

A brief summary is also provided at the end of this NCERT Solutions Class 10 Mathematics chapter 5 exercise 5.4, which will aid students in quickly memorising the full chapter as well as the formulas. NCERT Book Exercise 5.4 Class 10 Maths is an optional exercise with a total of 5 questions that may appear difficult at first and will necessitate some brainstorming.

Arithmetic Progressions Class 10 Chapter 5 Exercise: 5.4

Q1. Which term of the AP: is its first negative term $\small 121,117,113,...,$? [Hint : Find $n$ for $a_n<0$]

Answer:

Given AP is

$\small 121,117,113,...,$

Here $a = 121 \ and \ d = -4$

Let's suppose the nth term of the AP is the first negative term

Then,

$a_n = a+ (n-1)d$

If the nth term is negative, then $a_n < 0$

$\Rightarrow 121+(n-1)(-4) < 0$

$\Rightarrow 125<4n$

$\Rightarrow n > \frac{125}{4}=31.25$

Therefore, the first negative term must be the 32nd term

Q2. The sum of the third and the seventh terms of an AP is $\small 6$ and their product is $\small 8$ . Find the sum of first sixteen terms of the AP.

Answer:

It is given that the sum of the third and seventh terms of an AP is and their product is $\small 8$

$a_3= a+ 2d$

$a_7= a+ 6d$

Now,

$a_3+a_7= a+ 2d+a+6d= 6$

$\Rightarrow 2a+8d = 6$

$\Rightarrow a+4d = 3 \Rightarrow a = 3-4d \ \ \ \ \ \ \ \ \ \ \ \ -(i)$

And

$a_3.a_7 = (a+2d).(a+6d)=a^2+8ad +12d^2 = 8 \ \ \ \ \ \ \ -(ii)$

Putting the value from equation (i) in equation (ii), we will get

$\Rightarrow (3-4d)^2+8(3-4d)d+12d^2= 8$

$\Rightarrow 9+16d^2-24d+24d-32d^2+12d^2=8$

$\Rightarrow 4d^2 = 1$

$\Rightarrow d = \pm \frac{1}{2}$

Now,

Case (i) $d = \frac{1}{2}$

$a= 3 - 4 \times \frac{1}{2} = 1$

Then,

$S_{16}=\frac{16}{2}\left \{ 2\times 1+(16-1)\frac{1}{2} \right \}$

$S_{16}=76$

Case (ii) $d = -\frac{1}{2}$

$a= 3 - 4 \times \left ( -\frac{1}{2} \right ) = 5$

Then,

$S_{16}=\frac{16}{2}\left \{ 2\times 1+(16-1)\left ( -\frac{1}{2} \right ) \right \}$

$S_{16}=20$

Q3. A ladder has rungs $\small 25$ cm apart. (see Fig. $\small 5.7$ ). The rungs decrease uniformly in length from $\small 45$ cm at the bottom to $\small 25$ cm at the top. If the top and the bottom rungs are $\small 2\frac{1}{2}$ m apart, what is the length of the wood required for the rungs? [Hint:Number of rungs = $\frac{250}{25}+1$]

1635921645173

Answer:

It is given that

The total distance between the top and bottom rung $= 2\frac{1}{2}\ m = 250cm$

Distance between any two rungs = 25 cm

Total number of rungs = $\frac{250}{25}+1= 11$

And it is also given that the bottom-most rung is of 45 cm length and the top-most is of 25 cm length. As it is given that the length of rungs

decreases uniformly, it will form an AP with $a = 25, a_{11} = 45 \ and \ n = 11$

Now, we know that

$a_{11}= a+ 10d$

$\Rightarrow 45=25+10d$

$\Rightarrow d = 2$

Now, the total length of the wood required for the rungs is equal to

$S_{11} = \frac{11}{2}\left \{ 2\times 25+(11-1)2 \right \}$

$S_{11} = \frac{11}{2}\left \{ 50+20 \right \}$

$S_{11} = \frac{11}{2}\times 70$

$S_{11} =385 \ cm$

Therefore, the total length of the wood required for the rungs is equal to 385 cm

Q4 The houses of a row are numbered consecutively from $\ small1$ to $\ small49$. Show that there is a value of $\small x$ such that the sum of the numbers of the houses preceding the house numbered $\small x$ is equal to the sum of the numbers of the houses following it. Find the value of $\ small x$.

Answer:

It is given that the sum of the numbers of the houses preceding the house numbered $\small x$ is equal to the sum of the numbers of the

houses following it

And 1,2,3,.....,49 form an AP with a = 1 and d = 1

Now, we know that

$S_n= \frac{n}{2}\left \{ 2a+(n-1)d \right \}$

Suppose there exists an n term such that ( n < 49)

Now, according to the given conditions

The sum of the first n - 1 terms of AP = The sum of the terms following the nth term

Sum of the first n - 1 terms of AP = Sum of the whole AP - Sum of the first m terms of AP

i.e.

$S_{n-1}=S_{49}-S_n$

$\frac{n-1}{2}\left \{ 2a+((n-1)-1)d \right \}=\frac{49}{2}\left \{ 2a+(49-1)d \right \}-\frac{n}{2}\left \{ 2a+(n-1)d \right \}$

$\frac{n-1}{2}\left \{ 2+(n-2) \right \}=\frac{49}{2}\left \{ 2+48 \right \}-\frac{n}{2}\left \{ 2+(n-1) \right \}$

$\frac{n-1}{2}\left \{ n\right \}=\frac{49}{2}\left \{ 50 \right \}-\frac{n}{2}\left \{ n+1 \right \}$

$\frac{n^2}{2}-\frac{n}{2}=1225-\frac{n^2}{2}-\frac{n}{2}$

$n^2 = 1225$

$n = \pm 35$

Given House numbers are not negative,e so we reject n = -35

Therefore, the sum of no of houses preceding house no 35 is equal to the sum of no of houses following house number 35.

Q5. A small terrace at a football ground comprises of $\small 15$ steps each of which is $\small 50$ m long and built of solid concrete. Each step has a rise of $\small \frac{1}{4}\: m$ and a tread of $\small \frac{1}{2}\: m$ . (see Fig. $\small 5.8$ ). Calculate the total volume of concrete required to build the terrace.
[ Hint: Volume of concrete required to build the first step $\small =\frac{1}{4}\times \frac{1}{2}\times 50\: m^3$ ]

1635921665136

Answer:

It is given that

football ground comprises of $\small 15$ steps each of which is $\small 50$ m long and Each step has a rise of $\small \frac{1}{4}\: m$

and a tread of $\small \frac{1}{2}\: m$

Now,

The volume required to make the first step = $\frac{1}{4}\times \frac{1}{2}\times 50 = 6.25 \ m^3$

Similarly,

The volume required to make 2nd step = $\left ( \frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{1}{2}\times \frac{1}{2}\times 50 = 12.5 \ m^3$

And

The volume required to make 3rd step = $\left ( \frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{3}{4}\times \frac{1}{2}\times 50 = 18.75 \ m^3$

And so on

We can see that this is an AP with $a= 6.25 \ and \ d = 6.25$

Now, the total volume of concrete required to build the terrace of 15 such steps is

$S_{15} =\frac{15}{2}\left \{ 2 \times 6.25 +(15-1)6.25 \right \}$

$S_{15} =\frac{15}{2}\left \{ 12.5 +87.5\right \}$

$S_{15} =\frac{15}{2}\times 100$

$S_{15} =15\times 50 = 750$

Therefore, the total volume of concrete required to build the terrace of 15 such steps is $750 \ m^3$

Also Read

Topics Covered in Chapter 5, Arithmetic Progression: Exercise 5.4

  1. Finding the nth Term of an AP: The nth term of an AP can be determined using the formula (an) = a + (n - 1)d, where 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
  2. Solving Word Problems Using AP: These word problems are modelled using AP, by extracting the word problems and applying the appropriate formula to find the terms, the sum of related values.
  3. Finding the First Negative Term in an AP: In an AP, there is a common difference between given terms, and if there is a negative difference between given terms, then the smallest value for the 'n' term is negative.
  4. Using AP in Practical Calculations: AP is also used in various scenarios such as finding the distance, model pattern of growth, seating arrangement, business calculations, etc.
  5. Advanced-Level Questions: In this exercise, 5.4, many advanced-level questions are included that can not be solved directly by applying the AP formulas. To solve these problems, students need to understand the underlying concept and apply it accordingly.

Also see-

Frequently Asked Questions (FAQs)

Q: What is an Arithmetic progression?
A:

We can define Arithmetic Progression as a set of collective numbers where the difference between any two consecutive terms remains the same throughout the series. This difference between the two terms is called the common difference of the AP.

Q: What is the common difference of an Arithmetic progression?
A:

The differences between any two consecutive terms are the same which is known as the common difference of the arithmetic progression.

Q: Can common difference of an Arithmetic progression be negative?
A:

Yes, it the Arithmetic progression is a decreasing one. For example:

15, 12, 9, 6, 3

Q: Find the common difference of the arithmetic progression? 1, 4, 7, 10
A:

Common difference =7-4=3

Q: What is the formula for nth term of an AP
A:

nth term=a+(n-1)d

Where 'a' is the first term and 'd' is the common difference.

Q: What is the sum of first 10 natural numbers?
A:

First ten natural numbers are 1, 2, 3, .........,10

This is ap with a common difference of 1. 

Sum =0.5n(first term+ last term)

=0.5 x 10(1+10)=5 x11=55

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