**NCERT Solutions For Class 10 Maths Chapter 5 Arithmetic Progressions**

Class 10 Maths NCERT Solutions Chapter 5 Arithmetic Progressions – This Class 10 Maths Chapter 5 contains issues where the subsequent terms are derived by adding the same number to the preceding terms. The NCERT solutions for Class 10 Maths chapter 5 Arithmetic Progressions cover all of the class 10 exercise questions. Arithmetic Progression Class 10 solutions can aid you in both your academics and your future pursuits. These NCERT solutions for Class 10 provide a concise explanation for each question. While studying for the tests, you should consult the Arithmetic Progression Class 10 NCERT solutions.

NCERT Solutions for Class 10 Maths Chapter 6 Triangles

**NCERT Solutions for Class 10 Maths Chapter 5 ****Arithmetic Progressions Excercise: 5.1**

**Q1 (i) **In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (i) The taxi fare after each km when the fare is for the first km and for each additional .

**Answer:**

It is given that

Fare for = Rs. 15

And after that for each additional

Now,

Fare for = Fare of first km + Additional fare for 1 km

= Rs. 15 + 8 = Rs 23

Fare for = Fare of first km + Fare of additional second km + Fare of additional third km

= Rs. 23 + 8= Rs 31

Fare of n km =

( We multiplied by n – 1 because the first km was fixed and for rest, we are adding additional fare.

In this, each subsequent term is obtained by adding a fixed number (8) to the previous term.)

Now, we can clearly see that this is an A.P. with the first term (a) = 15 and common difference (d) = 8

**Q1 (ii) **In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (ii) The amount of air present in a cylinder when a vacuum pump removes of the air remaining in the cylinder at a time.

**Answer:**

It is given that

vacum pump removes of the air remaining in the cylinder at a time

Let us take initial quantity of air = 1

Now, the quantity of air removed in first step = 1/4

Remaining quantity after 1 ^{st }step

Similarly, Quantity removed after 2 ^{nd }step = Quantity removed in first step Remaining quantity after 1 ^{st }step

Now,

Remaining quantity after 2 ^{nd }step would be = Remaining quantity after 1 ^{st }step – Quantity removed after 2 ^{nd }step

Now, we can clearly see that

After the second step the difference between second and first and first and initial step is not the same, hence

the common difference (d) is not the same after every step

Therefore, it is not an AP

**Q1 (iii) **In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (iii) The cost of digging a well after every meter of digging, when it costs for the first metre and rises by for each subsequent meter.

**Answer:**

It is given that

Cost of digging of 1st meter = Rs 150

and

rises by for each subsequent meter

Therefore,

Cost of digging of first 2 meters = cost of digging of first meter + cost of digging additional meter

Cost of digging of first 2 meters = 150 + 50

= Rs 200

Similarly,

Cost of digging of first 3 meters = cost of digging of first 2 meters + cost of digging of additional meter

Cost of digging of first 3 meters = 200 + 50

= Rs 250

We can clearly see that 150, 200,250, … is in AP with each subsequent term is obtained by adding a fixed number (50) to the previous term.

Therefore, it is an AP with first term (a) = 150 and common difference (d) = 50

**Q1 (iv) **In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (iv) The amount of money in the account every year, when is deposited at compound interest at per annum .

**Answer:**

Amount in the beginning = Rs. 10000

Interest at the end of 1st year at the rate of

is of 10000 =

Therefore, amount at the end of 1st year will be

= 10000 + 800

= 10800

Now,

Interest at the end of 2nd year at rate of

is of 10800 =

Therefore,, amount at the end of 2 ^{nd }year

= 10800 + 864 = 11664

Since each subsequent term is not obtained by adding a unique number to the previous term; hence, it is not an AP

**Q2 (i) **Write first four terms of the AP, when the first term a and the common difference d are given as follows

**Answer:**

It is given that

Now,

Therefore, the first four terms of the given series are 10,20,30,40

**Q2 (ii) **Write first four terms of the AP when the first term a and the common difference d are given as follows: **Answer:**

It is given that

Now,

Therefore, the first four terms of the given series are -2,-2,-2,-2

**Q2 (iii) **Write first four terms of the AP when the first term a and the common difference d are given as follows

**Answer:**

It is given that

Now,

Therefore, the first four terms of the given series are 4,1,-2,-5

**Q2 (iv) **Write first four terms of the AP when the first term a and the common difference d are given as follows

**Answer:**

It is given that

Now,

Therefore, the first four terms of the given series are

**Q2 (v) **Write first four terms of the AP when the first term a and the common difference d are given as follows

**Answer:**

It is given that

Now,

Therefore, the first four terms of the given series are -1.25,-1.50,-1.75,-2

**Q3 (i) **For the following APs, write the first term and the common difference:

**Answer:**

Given AP series is

Now, first term of this AP series is **3**

Therefore,

First-term of AP series (a) = **3**

Now,

And common difference (d) =

Therefore, first term and common difference is **3 and -2 **respectively

**Q3 (ii) **For the following APs, write the first term and the common difference:

**Answer:**

Given AP series is

Now, the first term of this AP series is **-5**

Therefore,

First-term of AP series (a) = **-5**

Now,

And common difference (d) =

Therefore, the first term and the common difference is **-5 **and **4 **respectively

**Q3 (iii) **For the following APs, write the first term and the common difference:

**Answer:**

Given AP series is

Now, the first term of this AP series is

Therefore,

The first term of AP series (a) =

Now,

And common difference (d) =

Therefore, the first term and the common difference is and respectively

**Q3 (iv) **For the following APs, write the first term and the common difference:

**Answer:**

Given AP series is

Now, the first term of this AP series is **0.6**

Therefore,

First-term of AP series (a) = **0.6**

Now,

And common difference (d) =

Therefore, the first term and the common difference is **0.6 **and **1.1 **respectively.

**Q4 (i) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = **2**

Now,

We can clearly see that the difference between terms are not equal

Hence, given series is not an AP

**Q4 (ii) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

first term to this series is = **2**

Now,

We can clearly see that the difference between terms are equal and equal to

Hence, given series is in AP

Now, the next three terms are

Therefore, next three terms of given series are

**Q4 (iii) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = **-1.2**

Now,

We can clearly see that the difference between terms are equal and equal to **-2**

Hence, given series is in AP

Now, the next three terms are

Therefore, next three terms of given series are -9.2,-11.2,-13.2

**Q4 (iv) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = **-10**

Now,

We can clearly see that the difference between terms are equal and equal to **4**

Hence, given series is in AP

Now, the next three terms are

Therefore, next three terms of given series are 6,10,14

**Q4 (v) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = **3**

Now,

We can clearly see that the difference between terms are equal and equal to

Hence, given series is in AP

Now, the next three terms are

Therefore, next three terms of given series are

**Q4 (vi) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = 0.2

Now,

We can clearly see that the difference between terms are not equal

Hence, given series is not an AP

**Q4 (vii) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

first term to this series is = 0

Now,

We can clearly see that the difference between terms are equal and equal to **-4**

Hence, given series is in AP

Now, the next three terms are

Therefore, the next three terms of given series are -16,-20,-24

**Q4 (viii) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is =

Now,

We can clearly see that the difference between terms are equal and equal to **0**

Hence, given series is in AP

Now, the next three terms are

Therefore, the next three terms of given series are

**Q4 (ix) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = 1

Now,

We can clearly see that the difference between terms are not equal

Hence, given series is not an AP

**Q4 (x) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is = **a**

Now,

We can clearly see that the difference between terms are equal and equal to **a**

Hence, given series is in AP

Now, the next three terms are

Therefore, next three terms of given series are 5a,6a,7a

**Q4 (xi) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms. **Answer:**

Given series is

Now,

the first term to this series is = **a**

Now,

We can clearly see that the difference between terms are not equal

Hence, given series is not in AP

**Q4 (xii) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

We can rewrite it as

Now,

first term to this series is = **a**

Now,

We can clearly see that difference between terms are equal and equal to

Hence, given series is in AP

Now, the next three terms are

Therefore, next three terms of given series are

That is the next three terms are

**Q4 (xiii) **Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

Now,

the first term to this series is =

Now,

We can clearly see that the difference between terms are not equal

Hence, given series is not in AP

**Q4 (xiv) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

we can rewrite it as

Now,

the first term to this series is = **1**

Now,

We can clearly see that the difference between terms are not equal

Hence, given series is not in AP

**Q4 (xv) **Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.

**Answer:**

Given series is

we can rewrite it as

Now,

the first term to this series **is = 1**

Now,

We can clearly see that the difference between terms are equal and equal to **24**

Hence, given series is in AP

Now, the next three terms are

Therefore, the next three terms of given series are 97,121,145

**NCERT solutions for class 10 maths chapter 5 ****Arithmetic Progressions Excercise: 5.2**

**Q1 **Fill in the blanks in the following table, given that a is the first term, d the common difference and the th term of the AP:

a | d | n | ||

(i)(ii)(iii)(iv)(v) | 7 | 30 | 81018105 | 0 |

**Answer:**

(i)

It is given that

Now, we know that

Therefore,

(ii) It is given that

Now, we know that

(iii) It is given that

Now, we know that

Therefore,

(iv) It is given that

Now, we know that

Therefore,

(v) It is given that

Now, we know that

Therefore,

**Q2 (i) **Choose the correct choice in the following and justify: th term of the AP: is

**Answer:**

Given series is

Here,

and

Now, we know that

It is given that

Therefore,

Therefore, th term of the AP: is **-77**

Hence, Correct answer is (C)

**Q2 (ii) **Choose the correct choice in the following and justify : 11th term of the AP: is

(A) (B) (C) (D)

**Answer:**

Given series is

Here,

and

Now, we know that

It is given that

Therefore,

Therefore, 11th term of the AP: is **22**

Hence, the Correct answer is (B)

**Q3 (i) **In the following APs, find the missing terms in the boxes :

**Answer:**

Given AP series is

Here,

Now, we know that

Now,

Therefore, the missing term is **14**

**Q3 (ii) **In the following APs, find the missing terms in the boxes:

**Answer:**

Given AP series is

Here,

Now,

Now, we know that

Now,

And

Therefore, missing terms are **18 and 8**

AP series is 18,13,8,3

**Q3 (iii) **In the following APs, find the missing terms in the boxes :

**Answer:**

Given AP series is

Here,

Now, we know that

Now,

And

Therefore, missing terms are ** and 8**

AP series is

**Q3 (iv) **In the following APs, find the missing terms in the boxes :

**Answer:**

Given AP series is

Here,

Now, we know that

Now,

And

And

And

Therefore, missing terms are **-2,0,2,4**

AP series is -4,-2,0,2,4,6

**Q3 (v) **In the following APs, find the missing terms in the boxes :

**Answer:**

Given AP series is

Here,

Now,

Now, we know that

Now,

And

And

And

Therefore, missing terms are **53,23,8,-7**

AP series is 53,38,23,8,-7,-22

**Q4 **Which term of the AP : is ?**Answer:**

Given AP is

Let suppose that nth term of AP is 78

Here,

And

Now, we know that that

Therefore, value of 16th term of given AP is 78

**Q5 (i) **Find the number of terms in each of the following APs :

**Answer:**

Given AP series is

Let’s suppose there are n terms in given AP

Then,

And

Now, we know that

Therefore, there are **34 **terms in given AP

**Q5 (ii) **Find the number of terms in each of the following APs :

**Answer:**

Given AP series is

suppose there are n terms in given AP

Then,

And

Now, we know that

Therefore, there are **27 **terms in given AP

**Q6 **Check whether is a term of the AP :

**Answer:**

Given AP series is

Here,

And

Now,

suppose -150 is nth term of the given AP

Now, we know that

Value of n is not an integer

Therefore, -150 is not a term of AP

**Q7 **Find the st term of an AP whose th term is and the th term is .

**Answer:**

It is given that

th term of an AP is and the th term is

Now,

And

On solving equation (i) and (ii) we will get

Now,

Therefore, 31st terms of given AP is 178

**Q8 **An AP consists of terms of which rd term is and the last term is . Find the th term.**Answer:**

It is given that

AP consists of terms of which rd term is and the last term is

Now,

And

On solving equation (i) and (ii) we will get

Now,

Therefore, 29th term of given AP is 64

**Q9 **If the rd and the th terms of an AP are and respectively, which term of this AP is zero?**Answer:**

It is given that

rd and the th terms of an AP are and respectively

Now,

And

On solving equation (i) and (ii) we will get

Now,

Let nth term of given AP is 0

Then,

Therefore, 5th term of given AP is 0

**Q10 **The th term of an AP exceeds its th term by . Find the common difference.**Answer:**

It is given that

th term of an AP exceeds its th term by

i.e.

Therefore, the common difference of AP is 1

**Q11 **Which term of the AP : will be more than its th term?**Answer:**

Given AP is

Here,

And

Now, let’s suppose nth term of given AP is more than its th term

Then,

Therefore, 65th term of given AP is more than its th term

**Q12 **Two APs have the same common difference. The difference between their th terms is , what is the difference between their th terms?**Answer:**

It is given that

Two APs have the same common difference and difference between their th terms is

i.e.

Let common difference of both the AP’s is d

Now, difference between 1000th term is

Therefore, difference between 1000th term is 100

**Q 13 **How many three-digit numbers are divisible by ?**Answer:**

We know that the first three digit number divisible by 7 is 105 and last three-digit number divisible by 7 is 994

Therefore,

Let there are n three digit numbers divisible by 7

Now, we know that

Therefore, there are 128 three-digit numbers divisible by 7

**Q14 **How many multiples of lie between and ?**Answer:**

We know that the first number divisible by 4 between 10 to 250 is **12 **and last number divisible by 4 is **248**

Therefore,

Let there are n numbers divisible by 4

Now, we know that

Therefore, there are 60 numbers between 10 to 250 that are divisible by 4

**Q15 **For what value of , are the th terms of two APs: and equal?

**Answer:**

Given two AP’s are

and

Let first term and the common difference of two AP’s are a , a’ and d , d’

And

Now,

Let nth term of both the AP’s are equal

Therefore, the 13th term of both the AP’s are equal

**Q16 **Determine the AP whose third term is and the th term exceeds the th term by .**Answer:**

It is given that

3rd term of AP is and the th term exceeds the th term by

i.e.

And

Put the value of d in equation (i) we will get

Now, AP with first term = 4 and common difference = 6 is

4,10,16,22,…..

**Q17 **Find the th term from the last term of the AP : .

**Answer:**

Given AP is

Here,

And

Let suppose there are n terms in the AP

Now, we know that

So, there are 51 terms in the given AP and 20th term from the last will be 32th term from the starting

Therefore,

Therefore, 20th term from the of given AP is **158**

**Q18 **The sum of the th and th terms of an AP is and the sum of the th and th terms is Find the first three terms of the AP.**Answer:**

It is given that

sum of the th and th terms of an AP is and the sum of the th and th terms is

i.e.

And

On solving equation (i) and (ii) we will get

Therefore,first three of AP with a = -13 and d = 5 is

-13,-8,-3

**Q19 **Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?

**Answer:**

It is given that

Subba Rao started work at an annual salary of Rs 5000 and received an increment of Rs 200 each year

Therefore,

Let’s suppose after n years his salary will be Rs 7000

Now, we know that

Therefore, after **11 **years his salary will be Rs 7000

after 11 years, starting from 1995, his salary will reach to 7000, so we have to add 10 in 1995, because these numbers are in years

Thus , 1995+10 = 2005

**Q20 **Ramkali saved Rs 5 in the first week of a year and then increased her weekly savings by Rs . If in the th week, her weekly savings become Rs , find

**Answer:**

It is given that

Ramkali saved Rs 5 in the first week of a year and then increased her weekly savings by Rs

Therefore,

after th week, her weekly savings become Rs

Now, we know that

Therefore, after 10 weeks her saving will become Rs 20.75

**NCERT solutions for class 10 maths chapter 5 ****Arithmetic Progressions Excercise: 5.3**

**Q1 (i) **Find the sum of the following APs: to terms.

**Answer:**

Given AP is

to terms

Here,

And

Now, we know that

Therefore, the sum of AP to terms is **245**

**Q1 (ii) **Find the sum of the following APs: to terms.

**Answer:**

Given AP is

to terms.

Here,

And

Now, we know that

Therefore, the sum of AP to terms. is **-180**

**Q1 (iii) **Find the sum of the following APs: to terms.

**Answer:**

Given AP is

to terms..

Here,

And

Now, we know that

Therefore, the sum of AP to terms. is **5505**

**Q1 (iv) **Find the sum of the following APs: to terms.

**Answer:**

Given AP is

to terms.

Here,

And

Now, we know that

Therefore, the sum of AP to terms. is

**Q2 (i) **Find the sums given below :

**Answer:**

Given AP is

We first need to find the number of terms

Here,

And

Let suppose there are n terms in the AP

Now, we know that

Now, we know that

Therefore, the sum of AP is

**Q2 (ii) **Find the sums given below :

**Answer:**

Given AP is

We first need to find the number of terms

Here,

And

Let suppose there are n terms in the AP

Now, we know that

Now, we know that

Therefore, the sum of AP is **286**

**Q2 (iii) **Find the sums given below :

**Answer:**

Given AP is

We first need to find the number of terms

Here,

And

Let suppose there are n terms in the AP

Now, we know that

Now, we know that

Therefore, the sum of AP is **-8930**

**Q3 (i) **In an AP: given , , , find and .

**Answer:**

It is given that

Let suppose there are n terms in the AP

Now, we know that

Now, we know that

Therefore, the sum of the given AP is 440

**Q3 (ii) **In an AP: given , , find and .

**Answer:**

It is given that

Now, we know that

Therefore, the sum of given AP is **273**

**Q3 (iii) **In an AP: given find and .

**Answer:**

It is given that

Now, we know that

Therefore, the sum of given AP is **246**

**Q3 (iv) **In an AP: given find and **Answer:**

It is given that

Now, we know that

On solving equation (i) and (ii) we will get

Now,

Therefore, the value of d and 10th terms is -1 and 8 respectively

**Q3 (v) **In an AP: given , find and .

**Answer:**

It is given that

Now, we know that

Now,

**Q3 (vi) **In an AP: given find and .

**Answer:**

It is given that

Now, we know that

n can not be negative so the only the value of n is **5**

Now,

Therefore, value of n and nth term is 5 and 34 respectively

**Q3 (vii) **In an AP: given find and .

**Answer:**

It is given that

Now, we know that

Now, we know that

Now, put this value in (i) we will get

Therefore, value of n and d are respectively

**Q3 (viii) **In an AP: given find and .

**Answer:**

It is given that

Now, we know that

Now, we know that

Value of n cannot be negative so the only the value of n is **7**

Now, put this value in (i) we will get**a = -8**

Therefore, the value of n and a are **7 **and **-8 **respectively

**Q3 (ix) **In an AP: given find .

**Answer:**

It is given that

Now, we know that

Therefore, the value of d is 6

**Q3 (x) **In an AP: given and there are total terms. Find .

**Answer:**

It is given that

Now, we know that

Now, we know that

Therefore, the value of a is 4

**Q4 **How many terms of the AP: must be taken to give a sum of ?

**Answer:**

Given AP is

Here,

And

Now , we know that

Value of n can not be negative so the only the value of n is **12**

Therefore, the sum of **12 **terms of AP must be taken to give a sum of .

**Q5 **The first term of an AP is , the last term is and the sum is . Find the number of terms and the common difference.

**Answer:**

It is given that

Now, we know that

Now, we know that

Now, put this value in (i) we will get

Therefore, value of n and d are respectively

**Q6 **The first and the last terms of an AP are and respectively. If the common difference is , how many terms are there and what is their sum?

**Answer:**

It is given that

Now, we know that

Now, we know that

Therefore, there are **38 terms **and their sun is **6973**

**Q7 **Find the sum of first terms of an AP in which and nd term is .

**Answer:**

It is given that

Now, we know that

Now, we know that

Therefore, there are 22 **terms **and their sum is **1661**

**Q8 **Find the sum of first terms of an AP whose second and third terms are and respectively.

**Answer:**

It is given that

And

Now,

Now, we know that

Therefore, there are 51 **terms **and their sum is **5610**

**Q9 **If the sum of first terms of an AP is and that of terms is , find the sum of first terms.

**Answer:**

It is given that

Now, we know that

Similarly,

On solving equation (i) and (ii) we will get

a = 1 and d = 2

Now, the sum of first n terms is

Therefore, the sum of n terms is

**Q10 (i) **Show that form an AP where an is defined as below : Also find the sum of the first terms.

**Answer:**

It is given that

We will check values of for different values of n

and so on.

From the above, we can clearly see that this is an AP with the first term(a) equals to **7 **and common difference (d) equals to **4**

Now, we know that

Therefore, the sum of 15 terms is **525**

**Q10 (ii) **Show that form an AP where an is defined as below : . Also find the sum of the first terms in each case.

**Answer:**

It is given that

We will check values of for different values of n

and so on.

From the above, we can clearly see that this is an AP with the first term(a) equals to **4 **and common difference (d) equals to **-5**

Now, we know that

Therefore, the sum of 15 terms is **-465**

**Q11 **If the sum of the first terms of an AP is , what is the first term (that is )? What is the sum of first two terms? What is the second term? Similarly, find the rd, the th and the th terms

**Answer:**

It is given that

the sum of the first terms of an AP is

Now,

Now, first term is

Therefore, first term is **3**

Similarly,

Therefore, sum of first two terms is **4**

Now, we know that

Now,

Similarly,

**Q12 **Find the sum of the first positive integers divisible by .

**Answer:**

Positive integers divisible by 6 are

6,12,18,…

This is an AP with

Now, we know that

Therefore, sum of the first positive integers divisible by is **4920**

**Q13 **Find the sum of the first multiples of .

**Answer:**

First 15 multiples of 8 are

8,16,24,…

This is an AP with

Now, we know that

Therefore, sum of the first 15 multiple of 8 is **960**

**Q14 **Find the sum of the odd numbers between and .

**Answer:**

The odd number between 0 and 50 are

1,3,5,…49

This is an AP with

There are total **25 **odd number between 0 and 50

Now, we know that

Therefore, sum of the odd numbers between and ** 625**

**Q15 **A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs for the first day, Rs for the second day, Rs for the third day, etc., the penalty for each succeeding day being Rs more than for the preceding day. How much money the contractor has to pay a penalty, if he has delayed the work by days?

**Answer:**

It is given that

Penalty for delay of completion beyond a certain date is Rs for the first day, Rs for the second day, Rs for the third day and penalty for each succeeding day being Rs more than for the preceding day

We can clearly see that

200,250,300,….. is an AP with

Now, the penalty for 30 days is given by the expression

Therefore, the penalty for 30 days is **27750**

**Q16 **A sum of Rs is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs less than its preceding prize, find the value of each of the prizes.

**Answer:**

It is given that

Each price is decreased by 20 rupees,

Therefore, **d = -20 **and there are total 7 prizes so **n = 7 **and sum of prize money is Rs 700 so

Let a be the prize money given to the 1st student

Then,

Therefore, the prize given to the first student is **Rs 160**

Now,

Let is the prize money given to the next 6 students

then,

Therefore, prize money given to 1 to 7 student is **160,140,120,100,80,60.40**

**Q17 **In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I???? will plant tree, a section of Class II will plant trees and so on till Class XII. There are three sections in each class. How many trees will be planted by the students?

**Answer:**

First there are 12 classes and each class has 3 sections

Since each section of class 1 will plant 1 tree, so 3 trees will be planted by 3 sections of class 1. **Thus every class will plant 3 times the number of their class**

Similarly,

No. of trees planted by 3 sections of class 1 = 3

No. of trees planted by 3 sections of class 2 = 6

No. of trees planted by 3 sections of class 3 = 9

No. of trees planted by 3 sections of class 4 = 12

Its clearly an AP with first term **(a) **= **3 **and common difference **(d) **= **3 **and total number of classes **(n) **= **12**

Now, number of trees planted by 12 classes is given by

Therefore, number of trees planted by 12 classes is **234**

**Q18 **A spiral is made up of successive semicircles, with centres alternately at and ??????, starting with centre at , of radii as shown in Fig. . What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take )

[ **Hint **: Length of successive semicircles is with centres at respectively.]

**Answer:**

From the above-given figure

Circumference of 1st semicircle

Similarly,

Circumference of 2nd semicircle

Circumference of 3rd semicircle

It is clear that this is an AP with

Now, sum of length of 13 such semicircles is given by

Therefore, sum of length of 13 such semicircles is **143 cm**

**Q19 ** logs are stacked in the following manner: logs in the bottom row, in the next row, in the row next to it and so on (see Fig. ). In how many rows are the logs placed and how many logs are in the top row?

**Answer:**

As the rows are going up, the no of logs are decreasing,

We can clearly see that 20, 19, 18, …, is an AP.

and here

Let suppose 200 logs are arranged in ‘n’ rows,

Then,

Now,

case (i) **n = 25**

But number of rows can not be in negative numbers

Therefore, we will reject the value **n = 25case (ii) n = 16**

Therefore, the number of rows in which 200 logs are arranged is equal to

**5**

A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

[ **Hint **: To pick up the first potato and the second potato, the total distance (in metres) run by a competitor is ]

**Answer:**

Distance travelled by the competitor in picking and dropping 1st potato

Distance travelled by the competitor in picking and dropping 2nd potato

Distance travelled by the competitor in picking and dropping 3rd potato

and so on

we can clearly see that it is an AP with first term **(a) = 10 **and common difference **(d) = ****6**

There are 10 potatoes in the line

Therefore, total distance travelled by the competitor in picking and dropping potatoes is

Therefore, the total distance travelled by the competitor in picking and dropping potatoes is **370 m**

**NCERT solutions for class 10 maths chapter 5 ****Arithmetic Progressions Excercise: 5.4**

**Q1 **Which term of the AP: is its first negative term? [ **Hint **: Find for ]

**Answer:**

Given AP is

Here

Let suppose nth term of the AP is first negative term

Then,

If nth term is negative then **Answer:**

It is given that sum of third and seventh terms of an AP are and their product is

Now,

And

put value from equation (i) in (ii) we will get

Now,**case (i) **

Then,

case (ii)

Then,

**Q3 **A ladder has rungs cm apart. (see Fig. ). The rungs decrease uniformly in length from cm at the bottom to cm at the top. If the top and the bottom rungs are m apart, what is the length of the wood required for the rungs? [ **Hint: **Number of rungs

**Answer:**

It is given that

The total distance between the top and bottom rung

Distance between any two rungs = 25 cm

Total number of rungs =

And it is also given that bottom-most rungs is of 45 cm length and topmost is of 25 cm length.As it is given that the length of rungs decrease uniformly, it will form an AP with

Now, we know that

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

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 to . Show that there is a value of such that the sum of the numbers of the houses preceding the house numbered is equal to the sum of the numbers of the houses following it. Find this value of . [ **Hint **: ]

**Answer:**

It is given that the sum of the numbers of the houses preceding the house numbered 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

Suppose their exist an n term such that ( n < 49)

Now, according to given conditions

Sum of first n – 1 terms of AP = Sum of terms following the nth term

Sum of first n – 1 term of AP = Sum of whole AP – Sum of first m terms of AP

i.e.

Given House number are not negative so we reject n = -35

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

**Q5 **A small terrace at a football ground comprises of steps each of which is m long and built of solid concrete. Each step has a rise of and a tread of . (see Fig. ). Calculate the total volume of concrete required to build the terrace.

[ **Hint: **Volume of concrete required to build the first step ]

### Answer:

It is given that

football ground comprises of steps each of which is m long and Each step has a rise of and a tread of

Now,

The volume required to make the first step =

Similarly,

The volume required to make 2nd step =

And

The volume required to make 3rd step =

And so on

We can clearly see that this is an AP with

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

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