EXERCISE 5.2
Q 1, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Page No 105:
Question 1:
Fill in the blanks in the following table, given that a is the first term, d the common difference and an the nth term of the A.P.
a
|
d
|
n
|
an
| |
I
|
7
|
3
|
8
|
……
|
II
|
− 18
|
…..
|
10
|
0
|
III
|
…..
|
− 3
|
18
|
− 5
|
IV
|
− 18.9
|
2.5
|
…..
|
3.6
|
V
|
3.5
|
0
|
105
|
…..
|
Answer:
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 + (7) 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
18 = 9d
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 + (17) (−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
3.6 + 18.9 = (n − 1) 2.5
22.5 = (n − 1) 2.5
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 + 104 × 0
an = 3.5
Hence, an = 3.5
Q 2, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Page No 106:
Question 2:
Choose the correct choice in the following and justifyI. 30th term of the A.P: 10, 7, 4, …, is
A. 97 B. 77 C. − 77 D. − 87
II 11th term of the A.P.
isA. 28 B. 22 C. − 38 D.
Answer:
I. Given thatA.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 + (29) (−3)
a30 = 10 − 87 = −77
Hence, the correct answer is C.
II. Given that, A.P.
First term a = −3
Common difference, d = a2 − a1
We know that,
Hence, the answer is B.
Question 3:
In the following APs find the missing term in the boxesI.
II.
III.
IV.
V.
Answer:
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)
On subtracting (I) from (II), we obtain
−10 = 2d
d = −5
From equation (I), we obtain
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.,
We know that,
Therefore, the missing terms are
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 (1)
a6 = a + (6 − 1) d
−22 = a + 5d (2)
On subtracting equation (1) from (2), we obtain
− 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.
Q 4, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 4:
Which term of the A.P. 3, 8, 13, 18, … is 78?Answer:
3, 8, 13, 18, …For this A.P.,
a = 3
d = a2 − a1 = 8 − 3 = 5
Let nth term 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, 16th term of this A.P. is 78.
Q 5, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 5:
Find the number of terms in each of the following A.P.I. 7, 13, 19, …, 205
II.
Answer:
I. 7, 13, 19, …, 205For 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.,
Let there are n terms in this A.P.
Therefore, an = −47 and we know that,
Therefore, this given A.P. has 27 terms in it.
Q 6, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 6:
Check whether − 150 is a term of the A.P. 11, 8, 5, 2, …Answer:
For this A.P.,a = 11
d = a2 − a1 = 8 − 11 = −3
Let −150 be the nth term of this A.P.
We know that,
Clearly, n is not an integer.
Therefore, −150 is not a term of this A.P.
Q 7, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 7:
Find the 31st term of an A.P. whose 11th term is 38 and the 16th term is 73Answer:
Given that,a11 = 38
a16 = 73
We know that,
an = a + (n − 1) d
a11 = a + (11 − 1) d
38 = a + 10d (1)
Similarly,
a16 = a + (16 − 1) d
73 = a + 15d (2)
On subtracting (1) from (2), we obtain
35 = 5d
d = 7
From equation (1),
38 = a + 10 × (7)
38 − 70 = a
a = −32
a31 = a + (31 − 1) d
= − 32 + 30 (7)
= − 32 + 210
= 178
Hence, 31st term is 178.
Q 8, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 8:
An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th termAnswer:
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 obtain
94 = 47d
d = 2
From equation (I), we obtain
12 = a + 2 (2)
a = 12 − 4 = 8
a29 = a + (29 − 1) d
a29 = 8 + (28)2
a29 = 8 + 56 = 64
Therefore, 29th term is 64.
Q 9, Ex. 5.2 - Arithmetic Progressions (NCERT Maths Solutions Class 10th)
Question 9:
If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively. Which term of this A.P. is zero.Answer:
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 obtain
−12 = 6d
d = −2
From equation (I), we obtain
4 = a + 2 (−2)
4 = a − 4
a = 8
Let nth term 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, 5th term of this A.P. is 0.
Q 10, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 10:
If 17th term of an A.P. exceeds its 10th term by 7. Find the common difference.Answer:
We know that,For an A.P., an = a + (n − 1) d
a17 = a + (17 − 1) d
a17 = a + 16d
Similarly, a10 = a + 9d
It is given that
a17 − a10 = 7
(a + 16d) − (a + 9d) = 7
7d = 7
d = 1
Therefore, the common difference is 1.
Q 11, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 11:
Which term of the A.P. 3, 15, 27, 39, … will be 132 more than its 54th term?Answer:
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 nth term be 771.
an = a + (n − 1) d
771 = 3 + (n − 1) 12
768 = (n − 1) 12
(n − 1) = 64
n = 65
Therefore, 65th term was 132 more than 54th term.
Alternatively,
Let nth term be 132 more than 54th term.
Q 12, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 12:
Two APs have the same common difference. The difference between their 100th term is 100, what is the difference between their 1000th terms?Answer:
Let the first term of these A.P.s be a1 and a2 respectively 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
100th term of these A.P.s = 100
Therefore, (a1 + 99d) − (a2 + 99d) = 100
a1 − a2 = 100 (1)
Difference between 1000th terms of these A.P.s
(a1 + 999d) − (a2 + 999d) = a1 − a2
From equation (1),
This difference, a1 − a2 = 100
Hence, the difference between 1000th terms of these A.P. will be 100.
Q 13, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 13:
How many three digit numbers are divisible by 7Answer:
First three-digit number that is divisible by 7 = 105Next 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
889 = (n − 1) 7
(n − 1) = 127
n = 128
Therefore, 128 three-digit numbers are divisible by 7.
Q 14, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 14:
How many multiples of 4 lie between 10 and 250?Answer:
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 the nth term of this A.P.
Therefore, there are 60 multiples of 4 between 10 and 250.
Q 15, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 15:
For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10, 17, … equalAnswer:
63, 65, 67, …a = 63
d = a2 − a1 = 65 − 63 = 2
nth term of this A.P. = an = a + (n − 1) d
an= 63 + (n − 1) 2 = 63 + 2n − 2
an = 61 + 2n (1)
3, 10, 17, …
a = 3
d = a2 − a1 = 10 − 3 = 7
nth term of this A.P. = 3 + (n − 1) 7
an = 3 + 7n − 7
an = 7n − 4 (2)
It is given that, nth term of these A.P.s are equal to each other.
Equating both these equations, we obtain
61 + 2n = 7n − 4
61 + 4 = 5n
5n = 65
n = 13
Therefore, 13th terms of both these A.P.s are equal to each other.
Q 16, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Question 16:
Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12.Answer:
=a3 = 16a + (3 − 1) d = 16
a + 2d = 16 (1)
a7 − a5 = 12
[a+ (7 − 1) d] − [a + (5 − 1) d]= 12
(a + 6d) − (a + 4d) = 12
2d = 12
d = 6
From equation (1), we obtain
a + 2 (6) = 16
a + 12 = 16
a = 4
Therefore, A.P. will be
4, 10, 16, 22, …
Q 17, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
Page No 107:
Question 17:
Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253Answer:
Given A.P. is3, 8, 13, …, 253
Common difference for this A.P. is 5.
Therefore, this A.P. can be written in reverse order as
253, 248, 243, …, 13, 8, 3
For this 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, 20th term from the last term is 158.
Q - 18, Ex. 5.2, Arithmetic Progressions (NCERT Maths Solutions Class 10th)
Question 18:
Answer:
We know that,an = a + (n − 1) d
a4 = a + (4 − 1) d
a4 = a + 3d
Similarly,
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 (1)
a6 + a10 = 44
a + 5d + a + 9d = 44
2a + 14d = 44
a + 7d = 22 (2)
On subtracting equation (1) from (2), we obtain
2d = 22 − 12
2d = 10
d = 5
From equation (1), we obtain
a + 5d = 12
a + 5 (5) = 12
a + 25 = 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.
Q 19, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
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?Answer:
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 after nth year, 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.
Q 20, Ex 5.2 - Arithmetic Progressions - Chapter 5 - Maths Class 10th - NCERT
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 the nth week, her week, her weekly savings become Rs 20.75, find n.Answer:
Given that,a = 5
d = 1.75
an = 20.75
n = ?
an = a + (n − 1) d
n − 1 = 9
n = 10
Hence, n is 10.
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