# NCERT Solutions for Class 10 Maths Chapter 5 Exercise 5.2

## NCERT Solutions for Class 10 Maths Chapter 5 - Arithmetic Progressions Exercise 5.2

In this page we have NCERT Solutions for Class 10th Maths: Chapter 5 - Arithmetic Progressions for EXERCISE 5.2 . Hope you like them and do not forget to like , social_share and comment at the end of the page.
In an AP with first term a and common difference d, the nth term (or the general term) is given by
$a_n= a + (n - 1) d$
Question 1
Fill in the blanks in the following table, given that a is the first term,d the common difference and the nthterm of the A.P.

Formula to consider for solving these questions
an=a+ (n− 1)d
Where d -> common difference
A -> first term
n-> term
an -> nth term

(i)a= 7,d= 3,n= 8,an= ?
We know that,
For an A.P.an=a+ (n− 1)d
= 7 + (8 − 1) 3
= 7 + 21 = 28
Hence,an= 28

(ii) Given that
a= −18,n= 10,an= 0,d= ?
We know that,
an=a+ (n− 1)d
0 = − 18 + (10 − 1)d
d= 18/9 = 2
Hence, common difference,d= 2

(iii) Given that
d= −3,n= 18,an= −5
We know that,
an=a+ (n− 1)d
−5 =a+ (18 − 1) (−3)
−5 =a− 51
a= 51 − 5 = 46
Hence,a= 46

(iv)a= −18.9,d= 2.5,an= 3.6,n= ?
We know that,
an=a+ (n− 1)d
3.6 = − 18.9 + (n− 1) 2.5
22.5 = (n− 1) 2.5
(n- 1) = 22.5/2.5
n- 1 = 9
n= 10
Hence,n= 10

(v)a= 3.5,d= 0,n= 105,an= ?
We know that,
an=a+ (n− 1)d
an= 3.5 + (105 − 1) 0
an= 3.5
Hence,an= 3.5

Question 2
Choose the correct choice in the following and justify
(i) 30thterm of the A.P: 10, 7, 4, …, is
(A)97
(B)77
(C)−77
(D)−87

(ii) 11thterm of the A.P. -3, -1/2, ,2 .... is
(A) 28
(B) 22
(C) - 38
(D)-97/2

(i) Given that
A.P. 10, 7, 4, …
First term,a= 10
Common difference,d=a2−a1= 7 − 10 = −3
We know that,an=a+ (n− 1)d
a30= 10 + (30 − 1) (−3)
a30= 10 − 87 = −77
Hence, the correct answer is optionC.

(ii) Given that A.P. is-3, -1/2, ,2 ...
First term ,a= - 3
Common difference,d=a2−a1= (-1/2) - (-3)
= (-1/2)+ 3 = 5/2
We know that,an=a+ (n− 1)d
a11= 3+ (11 -1)(5/2)
a11= -3+ 25
a11= 22
Hence, the answer is option B.

Question 3
In the following APs find the missing term in the boxes.

Formula to consider for solving these questions

an=a+ (n− 1)d
Where d -> common difference
A -> first term
n-> term
an -> nth term

(i)For this A.P.,
a= 2
a3= 26
We know that,an=a+ (n− 1)d
a3= 2 + (3 - 1)d
26 = 2 + 2d
24 = 2d
d= 12
a2= 2 + (2 - 1) 12
= 14
Therefore, 14 is the missing term.

(ii)For this A.P.,
a2= 13 and
a4= 3
We know that,an=a+ (n− 1)d
a2=a+ (2 - 1)d
13 =a+d...(i)
a4=a+ (4 - 1)d
3 =a+ 3d...(ii)
Now solving these equations from the lesson of linear equations
On subtracting(i)from(ii), we get
- 10 = 2d
d= - 5
From equation(i), we get
13 =a+ (-5)
a= 18
a3= 18 + (3 - 1) (-5)
= 18 + 2 (-5) = 18 - 10 = 8
Therefore, the missing terms are 18 and 8 respectively.

(iii) For this A.P.,
a2= 13 and
a4= 3
We know that,an=a+ (n− 1)d
a2=a+ (2 - 1)d
13 =a+ d ...(i)
a4=a+ (4 - 1)d
3 =a+ 3d...(ii)
On subtracting(i)from(ii), we get,
- 10 = 2d
d = - 5
From equation(i), we get,
13 =a+ (-5)
a= 18
a3= 18 + (3 - 1) (-5)
= 18 + 2 (-5) = 18 - 10 = 8
Therefore, the missing terms are 18 and 8 respectively.

(iv) For this A.P.,
a= −4 and
a6= 6
We know that,
an=a+ (n− 1)d
a6= a + (6 − 1) d
6 = − 4 + 5d
10 = 5d
d= 2
a2=a+d= − 4 + 2 = −2
a3=a+ 2d= − 4 + 2 (2) = 0
a4=a+ 3d= − 4 + 3 (2) = 2
a5=a+ 4d= − 4 + 4 (2) = 4
Therefore, the missing terms are −2, 0, 2, and 4 respectively.

(v)
For this A.P.,
a2= 38
a6= −22
We know that
an=a+ (n− 1)d
a2=a+ (2 − 1)d
38 =a+d...(i)
a6=a+ (6 − 1)d
−22 =a+ 5d...(ii)
On subtracting equation(i)from(ii), we get
− 22 − 38 = 4d
−60 = 4d
d= −15
a=a2−d= 38 − (−15) = 53
a3=a+ 2d= 53 + 2 (−15) = 23
a4=a+ 3d= 53 + 3 (−15) = 8
a5=a+ 4d= 53 + 4 (−15) = −7
Therefore, the missing terms are 53, 23, 8, and −7 respectively.

Question 4
Which term of the A.P. 3, 8, 13, 18, … is 78?

We have
a= 3
d=a2−a1= 8 − 3 = 5
Let nthterm of this A.P. be 78.
an=a+ (n− 1)d
78 = 3 + (n− 1) 5
75 = (n− 1) 5
(n− 1) = 15
n= 16
Hence, 16thterm of this A.P. is 78.

Question 5
Find the number of terms in each of the following A.P.
(i) 7, 13, 19, …, 205
(ii) 18,31/2, 13,...., -47

(i) For this A.P.,
a= 7
d=a2−a1= 13 − 7 = 6
Let there are n terms in this A.P.
an= 205
We know that
an=a+ (n− 1)d
Therefore, 205 = 7 + (n− 1) 6
198 = (n− 1) 6
33 = (n− 1)
n= 34
Therefore, this given series has 34 terms in it.

(ii) For this A.P.,
a= 18

d=a2 – a1 = 31/2 - 18=-5/2

Let there are n terms in this A.P.
an= 205
an=a+ (n− 1)d
-47 = 18+ (n- 1) (-5/2)
-47 - 18 = (n- 1) (-5/2)
-65 = (n- 1)(-5/2)
(n- 1) = -130/-5
(n- 1) =26
n= 27
Therefore, this given A.P. has 27 terms in it.

Question 6
Check whether -150 is a term of the A.P. 11, 8, 5, 2, …
For this A.P.,
a= 11
d=a2−a1= 8 − 11 = −3
Let −150 be thenthterm of this A.P.
We know that,
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.
Therefore, - 150 is not a term of this A.P.

Question 7
Find the 31stterm of an A.P. whose 11thterm is 38 and the 16thterm is 73.

Given that,
a11= 38
a16= 73
We know that,
an=a+ (n− 1)d
a11=a+ (11 − 1)d
38 =a+ 10d...(i)
Similarly,
a16=a+ (16 − 1)d
73 =a+ 15d...(ii)
On subtracting(i)from(ii), we get
35 = 5d
d= 7
From equation(i),
38 =a+ 10 × (7)
38 − 70 =a
a= −32
Now that we know a and d of the A.P
a31=a+ (31 − 1)d
= − 32 + 30 (7)
= − 32 + 210
= 178
Hence, 31stterm is 178.

Question 8
An A.P. consists of 50 terms of which 3rdterm is 12 and the last term is 106. Find the 29thterm.

Given that,
a3= 12
a50= 106
We know that,
an=a+ (n− 1)d
a3=a+ (3 − 1)d
12 =a+ 2d...(i)
Similarly,a50=a+ (50 − 1)d
106 =a+ 49d...(ii)
On subtracting(i)from(ii), we get
94 = 47d
d= 2
From equation(i), we get
12 =a+ 2 (2)
a= 12 − 4 = 8
a29=a+ (29 − 1)d
a29= 8 + (28)2
a29= 8 + 56 = 64
Therefore, 29thterm is 64.

Question 9
If the 3rdand the 9thterms of an A.P. are 4 and − 8 respectively. Which term of this A.P. is zero.

Given that,
a3= 4
a9= −8
We know that,
an=a+ (n− 1)d
a3=a+ (3 − 1)d
4 =a+ 2d...(i)
a9=a+ (9 − 1)d
−8 =a+ 8d...(ii)
On subtracting equation(i)from(ii), we get,
−12 = 6d
d= −2
From equation(i), we get,
4 =a+ 2 (−2)
4 =a− 4
a= 8
Let nthterm of this A.P. be zero.
an=a+ (n− 1)d
0 = 8 + (n− 1) (−2)
0 = 8 − 2n+ 2
2n= 10
n= 5
Hence, 5thterm of this A.P. is 0.

Question 10
If 17thterm of an A.P. exceeds its 10thterm by 7. Find the common difference.

We know that,
For an A.P.,an=a+ (n− 1)d
So 17th term would be given as
a17=a+ (17 − 1)d
a17=a+ 16d
Similarly 10th term would be given as
a10=a+ 9d
It is given that in the question
a17−a10= 7
(a+ 16d) − (a+ 9d) = 7
7d= 7
d= 1
Therefore, the common difference is 1.

Question 11
Which term of the A.P. 3, 15, 27, 39, … will be 132 more than its 54thterm?

Given A.P. is 3, 15, 27, 39, …
a= 3
d=a2−a1= 15 − 3 = 12
a54=a+ (54 − 1)d
= 3 + (53) (12)
= 3 + 636 = 639
132 + 639 = 771
We have to find the term of this A.P. which is 771.
Let nthterm be 771.
an=a+ (n− 1)d
771 = 3 + (n− 1) 12
768 = (n− 1) 12
(n− 1) = 64
n= 65
Therefore, 65thterm was 132 more than 54thterm.

Question 12
Two APs have the same common difference. The difference between their 100thterm is 100, what is the difference between their 1000thterms?

Let the first term of these A.P.s be a1and a2respectively and the common difference of these A.P.s be d
For first A.P.,
a100=a1+ (100 − 1)d
=a1+ 99d
a1000=a1+ (1000 − 1)d
a1000=a1+ 999d
For second A.P.,
a100=a2+ (100 − 1)d
=a2+ 99d
a1000=a2+ (1000 − 1)d
=a2+ 999d
Given that, difference between
100thterm of these A.P.s = 100
Therefore, (a1+ 99d) − (a2+ 99d) = 100
a1−a2= 100 ...(i)
Difference between 1000thterms of these A.P.s
(a1+ 999d) − (a2+ 999d) =a1−a2
From equation(i),
This difference,a1−a2= 100
Hence, the difference between 1000thterms of these A.P. will be 100.

Question 13
How many three digit numbers are divisible by 7?

It is an interesting question which can be solved using arithmetic progression
First three-digit number that is divisible by 7 = 105
Next number = 105 + 7 = 112
Therefore, 105, 112, 119, …
All are three digit numbers which are divisible by 7 and thus, all these are terms of an A.P. having first term as 105 and common difference as 7.
The maximum possible three-digit number is 999. When we divide it by 7, the remainder will be 5. Clearly, 999 − 5 = 994 is the maximum possible three-digit number that is divisible by 7.
The series is as follows.
105, 112, 119, …, 994
Let 994 be the nth term of this A.P.
a= 105
d= 7
an= 994
n= ?
an=a+ (n− 1)d
994 = 105 + (n− 1) 7
(n− 1) = 127
n= 128
Therefore, 128 three-digit numbers are divisible by 7.

Question 14
How many multiples of 4 lie between 10 and 250?

It is an interesting question which can be solved using arithmetic progression.
First multiple of 4 that is greater than 10 is 12. Next will be 16.
Therefore, 12, 16, 20, 24, …
All these are divisible by 4 and thus, all these are terms of an A.P. with first term as 12 and common difference as 4.
When we divide 250 by 4, the remainder will be 2. Therefore, 250 − 2 = 248 is divisible by 4.
The series is as follows.
12, 16, 20, 24, …, 248
Let 248 be thenthterm of this A.P.
a= 12
d= 4
an=248
an=a+ (n- 1)d
248 = 12+ (n- 1) × 4
59 =n- 1
n= 60
Therefore, there are 60 multiples of 4 between 10 and 250.

Question 15
For what value of n, are thenthterms of two APs 63, 65, 67, and 3, 10, 17, … equal?

First series
63, 65, 67, …
a= 63
d=a2−a1= 65 − 63 = 2
nthterm of this A.P. =an=a+ (n− 1)d
an= 63 + (n− 1) 2 = 63 + 2n− 2
an= 61 + 2n...(i)

Second Series
3, 10, 17, …
a= 3
d=a2−a1= 10 − 3 = 7
nthterm of this A.P. = 3 + (n− 1) 7
an= 3 + 7n− 7
an= 7n− 4 ...(ii)
It is given that,nthterm of these A.P.s are equal to each other.
Equating both these equations, we obtain
61 + 2n= 7n− 4
5n= 65
n= 13
Therefore, 13thterms of both these A.P.s are equal to each other.

Question 16
Determine the A.P. whose third term is 16 and the 7thterm exceeds the 5thterm by 12.

We know that, nth term of the A.P is given by the expression
an=a+ (n− 1)d
Then 3th ,5th and 7th terms are
a3= 16
a+ (3 − 1)d= 16
a+ 2d= 16 ...(i)
a7= a+ (7 − 1)d= a+6d
a5= a+ (5 − 1)d=a+4d
Then
a7−a5= 12
[a+ 6d] − [a+ 4d]= 12
2d= 12
d= 6
From equation(i), we get,
a+ 2 (6) = 16
a+ 12 = 16
a= 4
Therefore, A.P. will be
4, 10, 16, 22, …

Question 17
Find the 20thterm from the last term of the A.P. 3, 8, 13, …, 253.

Given A.P. is
3, 8, 13, …, 253
d=5.
Therefore, this A.P. can be written in reverse order as
253, 248, 243, …, 13, 8, 5
For this reverse A.P.,
a= 253
d= 248 − 253 = −5
n= 20
a20=a+ (20 − 1)d
a20= 253 + (19) (−5)
a20= 253 − 95
a= 158
Therefore, 20thterm from the last term is 158.

Question 18
The sum of 4thand 8thterms of an A.P. is 24 and the sum of the 6thand 10thterms is 44. Find the first three terms of the A.P.

We know that, nth term of the A.P is given by the expression
an=a+ (n− 1)d
Then 4th term
a4=a+ (4 − 1)d
a4=a+ 3d
Similarly,8th,6th and 10th terms are
a8=a+ 7d
a6=a+ 5d
a10=a+ 9d
Given that,
a4+a8= 24
a+ 3d+a+ 7d= 24
2a+ 10d= 24
a+ 5d= 12 ...(i)

a6+a10= 44
a+ 5d+a+ 9d= 44
2a+ 14d= 44
a+ 7d= 22 ...(ii)
Equation (i) and (ii) are linear equation in two variable
On subtracting equation(i)from(ii), we get,
2d= 22 − 12
2d= 10
d= 5
From equation(i), we get
a+ 5d= 12
a+ 5 (5) = 12
a= −13
a2=a+d= − 13 + 5 = −8
a3=a2+d= − 8 + 5 = −3
Therefore, the first three terms of this A.P. are −13, −8, and −3.

Question 19
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?

It can be observed that the incomes that Subba Rao obtained in various years are in A.P. as every year, his salary is increased by Rs 200.
Therefore, the salaries of each year after 1995 are
5000, 5200, 5400, …
Here,a= 5000
d= 200
Let afternthyear, his salary be Rs 7000.
Therefore,an=a+ (n− 1)d
7000 = 5000 + (n− 1) 200
200(n− 1) = 2000
(n− 1) = 10
n= 11
Therefore, in 11th year, his salary will be Rs 7000.

Question 20
Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in thenthweek, her week, her weekly savings become Rs 20.75, find n.

Given that for the A.P
a= 5
d= 1.75
an= 20.75
We need to find out
n= ?
Now nth term is given by the expression
an=a+ (n− 1)d
20.75 = 5+ (n- 1) × 1.75
(n- 1) = 15.75/1.75 = 1575/175
= 63/7 = 9
n- 1 = 9
n= 10

Reference Books for class 10

Given below are the links of some of the reference books for class 10 math.

You can use above books for extra knowledge and practicing different questions.

### Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20

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