Exercise 5.2 (NCERT) – Arithmetic Progressions, Class 10
Below are the Quick links for all questions of Exercise 5.2, Arithmetic Progressions, Class 10. Click a link to view the solution of corresponding question.
| Chapter 5 | Exercise 5.2 | Class 10 |
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Below are the solutions of every question of exercise 5.2 of chapter 5, Arithmetic Progressions, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.
Question 1
Fill in the blanks in the following table, given that a is the first term, d the common difference and a the nth term of the AP:
(i) a=7, d=3, n=8, an=?
nth term of an AP
(ii) a=-18, d=?, n=10, an=0
(iii) a=?, d=-3, n=18, an=-5
(iv) a=-18.9, d=2.5, n=?, an=3.6
(v) a=3.5, d=0, n=105, an=?
Question 2
Choose the correct choice in the following and justify :
(i) 30th term of the AP: 10, 7, 4, . . . , is
(A) 97 (B) 77 (C) –77 (D) – 87
First Term, a = 10
Common Difference, d = 7 – 10 = -3
n = 30
nth terms of an AP
Answer: C
(ii) 11th term of the AP: – 3, -1/2, 2, … is
(A) 28 (B) 22 (C) -38 (D) -48 + 1/2
Answer: B
Question 3
In the following APs, find the missing terms in the boxes :
(i) 2, ? , 26
Given AP: 2, ? , 26
Middle term is the mean of adjacent terms in an AP series.
Answer: 14
(ii) ? , 13, ?, 3
Let us write the viven AP as : x, 13, y, 3
Middle term is the mean of adjacent terms in an AP series.
Now the series is : x, 13, 8, 3
Now 13 is mean of x and 8.
Answer: 18, 8
(iii) 5, ?, ?, 19/2
First term, a = 5
Let the common difference = d
nth terms of an AP
Let us now find the missing terms
Answer: 13/2, 8
(iv) -4, ?, ?, ?, ?, 6
First term, a = -4
Let the common difference = d
nth terms of an AP
Let us now find the missing terms
Answer: -2, 0, 2, 4
(v) ?, 38, ?, ?, ?, -22
Let the First term = a
Let the common difference = d
nth terms of an AP
Subtract (2) from (1)
Substitute value of d in (2)
Let us now find the missing terms
Answer: 53, 23, 8, -7
Question 4
Which term of the AP : 3, 8, 13, 18, . . . ,is 78?
First term (a) = 3
Common difference (d) = 8 – 3 = 5
nth terms of an AP
Answer: 16th term
Question 5
Find the number of terms in each of the following APs :
(i) 7, 13, 19, . . . , 205
First term (a) = 7
Common difference (d) = a₂ – a₁ = 13 – 7 = 6
nth terms of an AP
This means
Answer: n = 34
(ii) 18, 15+1/2, 13, . . . , – 47
First term (a) = 18
Common difference
nth terms of an AP
This means
Answer: n = 27
Question 6
Check whether – 150 is a term of the AP : 11, 8, 5, 2 . . .
First term (a) = 11
Common difference (d) = a₂ – a₁ = 8 – 11 = -3
nth terms of an AP
This means
Since n is not an integer this implies -150 is not a term of the AP.
Answer: -150 is not a term of given AP
Question 7
Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
nth terms of an AP
This means
Subtract (1) from (2)
Substitute value of d in (1)
Let’s find 31st term.
Answer: a31 = 178
Question 8
An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
nth terms of an AP
This means
Subtract (1) from (2)
Substitute value of d in (1)
Let’s find 29th term.
Answer: a29 = 64
Question 9
If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
nth terms of an AP
This means
Subtract (1) from (2)
Substitute value of d in (1)
Let nth term is 0, this implies
Answer: 5th term
Question 10
The 17th term of an AP exceeds its 10th term by 7. Find the common difference.
nth terms of an AP
This means
Answer: d = 1
Question 11
Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
nth terms of an AP
Also given
Let kth term is 132 more than 54th term, this implies
Answer: 65th term
Question 12
Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?
Let the two APs be
First AP: a1, a2, a3, …
Second AP: b1, b2, b3, …
Let Common difference for both APs = d
nth terms of the APs will be
Difference between 100th terms = 100
Let’s find difference between 1000th terms
Answer: 100
Question 13
How many three-digit numbers are divisible by 7?
Three-digit numbers range from 100 to 999
The smallest 3-digit number divisible by 7 = 105
The largest 3-digit number divisible by 7 = 994
All 3-digit numbers divisible by 7 are:
105, 112, 119, ….. , 994
This is an AP series.
First term, a = 105
Common Difference, d = 7
Let the number of terms in this AP = n
We know that, nth terms of an AP
This implies
Answer: 128
Question 14
How many multiples of 4 lie between 10 and 250?
Smallest multiple of 4 > 10 = 12
Largest multiple of 4 < 250 = 248
All multiples of 4 between numbers between 10 and 250 are:
12, 16, 20, …., 248
This is an AP series.
First term, a = 12
Common Difference, d = 4
Let the number of terms in this AP = n
We know that, nth terms of an AP
This implies
Answer: 60
Question 15
For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?
First AP: 63, 65, 67, …
Second AP: 3, 10, 17, …
nth Terms of both APs are equal, this implies
Answer: 13
Question 16
Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
Let the AP be
a, a + d, a + 2d, a + 3d,…
3rd term = 16
7th term – 5th term = 12
Substitute the value of d in (1)
AP series:
a, a + d, a + 2d, a + 3d,…
4, 10, 16, 22, …
Answer: 4, 10, 16, 22, …
Question 17
Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.
Given AP
3, 8, 13,…, 253
a = 3, d = 5, l = 253
nth terms of an AP
Let us find n for for last term
20th term from the last = (51 – 20 + 1)th term from start = 32nd term from start
Answer: 158
Question 18
The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
Let the AP be:
a, a + d, a + 2d,…
Sum of 4th and 8th terms = 24
Sum of 6th and 10th terms = 44
Subtract (1) from (2)
Substitute value of d in (1)
AP series:
a, a + d, a + 2d, …
-13, -8, -3, …
Answer: -13, -8, -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?
Salary in 1995 = 5000
Salary in 1996 = 5000 + 200 = 5200
Salary in 1997 = 5200 + 200 = 5400
and so on …
5000, 5200, 5400, … is an AP series
First term, a = 5200
Common difference, d = 200
Consider nth term will be 7000
n = 11 means in 11th year the salary will be 7000.
=> 1995 + 11 – 1 = 2005
In year 2005, the salary will be Rs. 7000.
Answer: 11th years, 2005
Why 2005 and not 2006? Watch the video.
Question 20
Ramkali saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs. 20.75, find n.
Savings in the 1st week = Rs. 5
Savings in the 2nd week = 5 + 1.75 = Rs. 6.75
Savings in the 3rd week = 6.75 + 1.75 = Rs. 8.5
Savings in the 4th week = 8.5 + 1.75 = Rs. 10.25
and so on…
5, 6.75, 8.5, 10.25, … is an AP series
First term, a = 5
Common difference, d = 1.75
Savings in the nth week are Rs. 20.75.
Answer: 10