Mathematics · Textbook solutions

Arithmetic Progressions

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 114 questions

5.2 Arithmetic Progressions

30 q

Solved Examples

Worked · 2

5.1 Eg.1 $\dfrac{3}{2}$
5.1 Eg.2 $-1$

Exercise 5.1

Practice · 28

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
Ex 5.1 Q1 (i) $The taxi fare after each km when the fare is ₹ 15 for the first km and ₹ 8 for each additional km.$
Ex 5.1 Q1 (ii) $\dfrac{1}{4}$
Ex 5.1 Q1 (iii) $The cost of digging a well after every metre of digging, when it costs ₹ 150 for the first metre and rises by ₹ 50 for each subsequent metre.$
Ex 5.1 Q1 (iv) $The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8 % per annum.$
Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows:
Ex 5.1 Q2 (i) $a = 10$
Ex 5.1 Q2 (ii) $a = -2$
Ex 5.1 Q2 (iii) $a = 4$
Ex 5.1 Q2 (iv) $a = -1$
Ex 5.1 Q2 (v) $a = -1.25$
For the following APs, write the first term and the common difference:
Ex 5.1 Q3 (i) $-1$
Ex 5.1 Q3 (ii) $-5$
Ex 5.1 Q3 (iii) $\dfrac{1}{3}$
Ex 5.1 Q3 (iv) $0.6, 1.7, 2.8, 3.9, . . .$
Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms.
Ex 5.1 Q4 (i) $2, 4, 8, 16, . . .$
Ex 5.1 Q4 (ii) $\dfrac{5}{2}$
Ex 5.1 Q4 (iii) $-1.2$
Ex 5.1 Q4 (iv) $-10$
Ex 5.1 Q4 (v) $3 + \sqrt{2}$
Ex 5.1 Q4 (vi) $0.2, 0.22, 0.222, 0.2222, . . .$
Ex 5.1 Q4 (vii) $-4$
Ex 5.1 Q4 (viii) $-\dfrac{1}{2}$
Ex 5.1 Q4 (ix) $1, 3, 9, 27, . . .$
Ex 5.1 Q4 (x) $a$
Ex 5.1 Q4 (xi) $a$
Ex 5.1 Q4 (xii) $\sqrt{2}$
Ex 5.1 Q4 (xiii) $\sqrt{3}$
Ex 5.1 Q4 (xiv) $1^2$
Ex 5.1 Q4 (xv) $1^2$

5.3 nth Term of an AP

38 q

Solved Examples

Worked · 8

5.2 Eg.3 $Find the 10th term of the AP : 2, 7, 12, . . .$
5.2 Eg.4 $-81$
5.2 Eg.5 $Determine the AP whose 3rd term is 5 and the 7th term is 9.$
5.2 Eg.6 $Check whether 301 is a term of the list of numbers 5, 11, 17, 23, . . .$
5.2 Eg.7 $How many two-digit numbers are divisible by 3?$
5.2 Eg.8 $-62$
5.2 Eg.9 $A sum of ₹ 1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of 30 years making use of this fact.$
5.2 Eg.10 $In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?$

Exercise 5.2

Practice · 30

Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$ th term of the AP:
Ex 5.2 Q1 (i) $a$
Ex 5.2 Q1 (ii) $a$
Ex 5.2 Q1 (iii) $a$
Ex 5.2 Q1 (iv) $a$
Ex 5.2 Q1 (v) $a$
Choose the correct choice in the following and justify :
Ex 5.2 Q2 (i) $30th term of the AP: 10, 7, 4, . . . , is$
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Ex 5.2 Q2 (ii) $-3$
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In the following APs, find the missing terms in the boxes :
Ex 5.2 Q3 (i) $\square$
Ex 5.2 Q3 (ii) $\square$
Ex 5.2 Q3 (iii) $\square$
Ex 5.2 Q3 (iv) $-4$
Ex 5.2 Q3 (v) $\square$
Ex 5.2 Q4 $Which term of the AP : 3, 8, 13, 18, . . . , is 78?$
Ex 5.2 Q6 $-150$
Ex 5.2 Q7 $Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.$
Ex 5.2 Q8 $An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.$
Ex 5.2 Q9 $-8$
Ex 5.2 Q10 $The 17th term of an AP exceeds its 10th term by 7. Find the common difference.$
Ex 5.2 Q11 $Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?$
Ex 5.2 Q12 $Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?$
Ex 5.2 Q13 $How many three-digit numbers are divisible by 7?$
Ex 5.2 Q14 $How many multiples of 4 lie between 10 and 250?$
Ex 5.2 Q15 $n$
Ex 5.2 Q16 $Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.$
Ex 5.2 Q17 $Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.$
Ex 5.2 Q18 $The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.$
Ex 5.2 Q19 $Subba Rao started work in 1995 at an annual salary of ₹ 5000 and received an increment of ₹ 200 each year. In which year did his income reach ₹ 7000?$
Ex 5.2 Q20 $n$
Find the number of terms in each of the following APs :
Ex 5.2 Q5 (i) $7, 13, 19, . . . , 205$
Ex 5.2 Q5 (ii) $15\dfrac{1}{2}$

$a$	$d$	$n$	$a_n$
7	3	8	. . .

$a$	$d$	$n$	$a_n$
$-18$	. . .	10	0

$a$	$d$	$n$	$a_n$
. . .	$-3$	18	$-5$

$a$	$d$	$n$	$a_n$
$-18.9$	2.5	. . .	3.6

$a$	$d$	$n$	$a_n$
3.5	0	105	. . .

5.4 Sum of First n Terms of an AP

41 q

Solved Examples

Worked · 6

5.3 Eg.11 $-2$
5.3 Eg.12 $If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.$
5.3 Eg.13 $How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78?$
5.3 Eg.14 $n$
5.3 Eg.15 $n$
5.3 Eg.16 $A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find : (i) the production in the 1st year (ii) the production in the 10th year (iii) the total production in first 7 years$

Exercise 5.3

Practice · 35

Find the sum of the following APs:
Ex 5.3 Q1 (i) $2, 7, 12, . . ., to 10 terms.$
Ex 5.3 Q1 (ii) $-37$
Ex 5.3 Q1 (iii) $0.6, 1.7, 2.8, . . ., to 100 terms.$
Ex 5.3 Q1 (iv) $\dfrac{1}{15}$
Find the sums given below :
Ex 5.3 Q2 (i) $7 + 10\dfrac{1}{2} + 14 + \ldots + 84$
Ex 5.3 Q2 (ii) $34 + 32 + 30 + \ldots + 10$
Ex 5.3 Q2 (iii) $-5 + (-8) + (-11) + \ldots + (-230)$
In an AP:
Ex 5.3 Q3 (i) $a = 5$
Ex 5.3 Q3 (ii) $a = 7$
Ex 5.3 Q3 (iii) $a_{12} = 37$
Ex 5.3 Q3 (iv) $a_3 = 15$
Ex 5.3 Q3 (v) $d = 5$
Ex 5.3 Q3 (vi) $a = 2$
Ex 5.3 Q3 (vii) $a = 8$
Ex 5.3 Q3 (viii) $a_n = 4$
Ex 5.3 Q3 (ix) $a = 3$
Ex 5.3 Q3 (x) $l = 28$
Ex 5.3 Q4 $How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?$
Ex 5.3 Q5 $The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.$
Ex 5.3 Q6 $The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?$
Ex 5.3 Q7 $d = 7$
Ex 5.3 Q8 $Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.$
Ex 5.3 Q9 $n$
Ex 5.3 Q12 $Find the sum of the first 40 positive integers divisible by 6.$
Ex 5.3 Q13 $Find the sum of the first 15 multiples of 8.$
Ex 5.3 Q14 $Find the sum of the odd numbers between 0 and 50.$
Ex 5.3 Q15 $A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc., the penalty for each succeeding day being ₹ 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?$
Ex 5.3 Q16 $A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.$
Ex 5.3 Q17 $In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?$
Ex 5.3 Q18 $\pi = \dfrac{22}{7}$
Ex 5.3 Q19 $200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?$
Ex 5.3 Q20 $2 \times 5 + 2 \times (5 + 3)$
Show that $a_1$ , $a_2$ , . . ., $a_n$ , . . . form an AP where $a_n$ is defined as below. Also find the sum of the first 15 terms in each case.
Ex 5.3 Q10 (i) $a_n = 3 + 4n$
Ex 5.3 Q10 (ii) $a_n = 9 - 5n$
Ex 5.3 Q11 $n$

Exercise 5.4 (Optional)

5 q

Ex 5.4 Q1 $n$
Ex 5.4 Q2 $The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.$
Ex 5.4 Q3 $2\dfrac{1}{2}$
Ex 5.4 Q4 $x$
Ex 5.4 Q5 $\dfrac{1}{4}$

What the board has asked from this chapter

Questions per paper · 67 papers · 2022 to 2026

2026

31 q

1 $The number of multiples of 6 lying between 25 and 363 is :$
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nth Term of an AP
3 $n^{\text{th}}$
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nth Term of an AP
3 $a_{14} - a_8 = 24$
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nth Term of an AP
5 $2x$
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Arithmetic Progressions and Common Difference
6 $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \ldots$
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Arithmetic Progressions and Common Difference
6 $The number of multiples of 4 lying between 12 and 250 is :$
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nth Term of an AP
7 $Which of the following sequence is not an A.P. ?$
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Arithmetic Progressions and Common Difference
7 $p$
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nth Term of an AP
8 $a_n$
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nth Term of an AP
11 $n^{\text{th}}$
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nth Term of an AP
16 $If sum of first ten terms of an A.P. is zero with a as the first term and d, the common difference, which of the following relation is true ?$
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Sum of First n Terms of an AP
17 $a = -3$
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Sum of First n Terms of an AP
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the options (A), (B), (C) and (D) as given below.
20 $\dfrac{n - 1}{2}$
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Sum of First n Terms of an AP
22 (a) $-10$
Sum of First n Terms of an AP
22 (b) $n^{\text{th}}$
Sum of First n Terms of an AP
23 (a) $In an A.P., the first term is 4 and the last term is 31. If sum of all the terms is 175, find the number of terms and the common difference.$
Sum of First n Terms of an AP
23 (b) $How many terms of the A.P. 21, 18, 15, .... must be added to get the sum zero ?$
Sum of First n Terms of an AP
27 (a) $15^{\text{th}}$
Sum of First n Terms of an AP
27 (b) $2n^2 + 13n$
Sum of First n Terms of an AP
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato. The other potatoes are arranged 3 m apart in a straight line, with a total of 10 potatoes, as shown in the figure : A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions :
36 (i) $What is the distance covered to pick up the first potato and drop it in bucket ?$
Arithmetic Progressions and Common Difference
Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹ 1,18,000 by paying every month, starting with the first instalment of ₹ 1,000 and he increases the instalment by ₹ 100 every month. Based on the information given above, answer the following questions :
36 (i) $30^{\text{th}}$
nth Term of an AP
Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹ 1,18,000 by paying every month, starting with the first instalment of ₹ 1,000 and he increases the instalment by ₹ 100 every month. Based on the information given above, answer the following questions :
36 (ii) $If the total number of instalments is 40, what is the amount paid in the last instalment ?$
nth Term of an AP
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato. The other potatoes are arranged 3 m apart in a straight line, with a total of 10 potatoes, as shown in the figure : A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions :
36 (ii) $What is the distance covered to pick up the second potato and drop it in bucket ?$
Arithmetic Progressions and Common Difference
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato. The other potatoes are arranged 3 m apart in a straight line, with a total of 10 potatoes, as shown in the figure : A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions :
36 (iii) (a) $What is the total distance the competitor has to run ?$
Sum of First n Terms of an AP
Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹ 1,18,000 by paying every month, starting with the first instalment of ₹ 1,000 and he increases the instalment by ₹ 100 every month. Based on the information given above, answer the following questions :
36 (iii) (a) $30^{\text{th}}$
Sum of First n Terms of an AP
Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹ 1,18,000 by paying every month, starting with the first instalment of ₹ 1,000 and he increases the instalment by ₹ 100 every month. Based on the information given above, answer the following questions :
36 (iii) (b) $Find the ratio of the tenth instalment to the last instalment.$
nth Term of an AP
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato. The other potatoes are arranged 3 m apart in a straight line, with a total of 10 potatoes, as shown in the figure : A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions :
36 (iii) (b) $If average speed of competitor is 5 m/s, then find the average time taken by competitor to put all the potatoes in the bucket.$
Sum of First n Terms of an AP
'Kolam' is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below. Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on. Based on the above, answer the following questions :
37 (i) $Show that number of dots given above form an A.P. Write the first term and common difference.$
Arithmetic Progressions and Common Difference
'Kolam' is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below. Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on. Based on the above, answer the following questions :
37 (ii) $n^{\text{th}}$
nth Term of an AP
'Kolam' is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below. Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on. Based on the above, answer the following questions :
37 (iii) (a) $The pattern is expanded on a large ground. If total 220 dots are used, then find the number of squares formed.$
Sum of First n Terms of an AP
'Kolam' is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below. Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on. Based on the above, answer the following questions :
37 (iii) (b) $Is it possible to complete n number of squares using 100 dots ? If yes, then find the value of n.$
Sum of First n Terms of an AP

2025

29 q

2 $9^{\text{th}}$
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nth Term of an AP
3 $Three numbers in AP have the sum 30. What is its middle term ?$
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Arithmetic Progressions and Common Difference
11 $2m^2 + 3m$
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Sum of First n Terms of an AP
15 $10^{\text{th}}$
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nth Term of an AP
17 $11^{\text{th}}$
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nth Term of an AP
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given below.
19 $5, 1, -3, -7, \ldots$
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Arithmetic Progressions and Common Difference
Directions : In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from following :
20 $10^{\text{th}}$
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nth Term of an AP
27 $Find the sum of all 3-digit natural numbers which are divisible by 11.$
Sum of First n Terms of an AP
31 $1^{\text{st}}$
nth Term of an AP
32 (a) $n^{\text{th}}$
Sum of First n Terms of an AP
32 (b) $10^{\text{th}}$
Sum of First n Terms of an AP
33 (a) $The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.$
Sum of First n Terms of an AP
33 (b) $The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.$
Sum of First n Terms of an AP
A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10. Based on the information given above, answer the following questions :
36 (i) $Write the fourth, fifth and sixth term of the Arithmetic Progression so formed.$
Arithmetic Progressions and Common Difference
A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10. Based on the information given above, answer the following questions :
36 (ii) $8^{\text{th}}$
nth Term of an AP
A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10. Based on the information given above, answer the following questions :
36 (iii) (a) $Find the total distance run after completing all 10 rounds.$
Sum of First n Terms of an AP
A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10. Based on the information given above, answer the following questions :
36 (iii) (b) $If a runner completes only the first 6 rounds, what is the total distance run by the runner ?$
Sum of First n Terms of an AP
Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of cable car ride from base point to top most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which car moves. The distance of first pole from base point is 200 m and subsequent poles are installed at equal interval of 150 m. Further, the distance of last pole from the top is 300 m. Based on above information, answer the following questions using Arithmetic Progression :
37 (i) $10^{\text{th}}$
nth Term of an AP
In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on. Based on given information, answer the following questions using Arithmetic Progression.
37 (i) $How many triangles will be there in bottom most row ?$
nth Term of an AP
In order to organise, Annual Sports Day, a school prepared an eight lane running track with an integrated football field inside the track area as shown below : The length of innermost lane of the track is 400 m and each subsequent lane is 7.6 m longer than the preceding lane. Based on given information, answer the following questions, using concept of Arithmetic Progression.
37 (i) $6^{\text{th}}$
nth Term of an AP
In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on. Based on given information, answer the following questions using Arithmetic Progression.
37 (ii) $How many triangles will be there in fourth row from the bottom ?$
nth Term of an AP
In order to organise, Annual Sports Day, a school prepared an eight lane running track with an integrated football field inside the track area as shown below : The length of innermost lane of the track is 400 m and each subsequent lane is 7.6 m longer than the preceding lane. Based on given information, answer the following questions, using concept of Arithmetic Progression.
37 (ii) $8^{\text{th}}$
nth Term of an AP
Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of cable car ride from base point to top most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which car moves. The distance of first pole from base point is 200 m and subsequent poles are installed at equal interval of 150 m. Further, the distance of last pole from the top is 300 m. Based on above information, answer the following questions using Arithmetic Progression :
37 (ii) $15^{\text{th}}$
nth Term of an AP
Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of cable car ride from base point to top most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which car moves. The distance of first pole from base point is 200 m and subsequent poles are installed at equal interval of 150 m. Further, the distance of last pole from the top is 300 m. Based on above information, answer the following questions using Arithmetic Progression :
37 (iii) (a) $15^{\text{th}}$
nth Term of an AP
In order to organise, Annual Sports Day, a school prepared an eight lane running track with an integrated football field inside the track area as shown below : The length of innermost lane of the track is 400 m and each subsequent lane is 7.6 m longer than the preceding lane. Based on given information, answer the following questions, using concept of Arithmetic Progression.
37 (iii) (a) $While practicing for a race, a student took one round each in first six lanes. Find the total distance covered by the student.$
Sum of First n Terms of an AP
In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on. Based on given information, answer the following questions using Arithmetic Progression.
37 (iii) (a) $8^{\text{th}}$
Sum of First n Terms of an AP
In order to organise, Annual Sports Day, a school prepared an eight lane running track with an integrated football field inside the track area as shown below : The length of innermost lane of the track is 400 m and each subsequent lane is 7.6 m longer than the preceding lane. Based on given information, answer the following questions, using concept of Arithmetic Progression.
37 (iii) (b) $A student took one round each in lane 4 to lane 8. Find the total distance covered by the student.$
Sum of First n Terms of an AP
Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of cable car ride from base point to top most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which car moves. The distance of first pole from base point is 200 m and subsequent poles are installed at equal interval of 150 m. Further, the distance of last pole from the top is 300 m. Based on above information, answer the following questions using Arithmetic Progression :
37 (iii) (b) $Find the total number of poles installed along the entire journey.$
nth Term of an AP
In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on. Based on given information, answer the following questions using Arithmetic Progression.
37 (iii) (b) $5^{\text{th}}$
Sum of First n Terms of an AP

2024

28 q

1 $(4^{\text{th}})$
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nth Term of an AP
2 $\dfrac{1}{2x}, \dfrac{1 - 4x}{2x}, \dfrac{1 - 8x}{2x}, \ldots$
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Arithmetic Progressions and Common Difference
2 $(a) = -16$
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Sum of First n Terms of an AP
3 $a_{15} - a_{11} = 48$
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nth Term of an AP
3 $(4^{\text{th}})$
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nth Term of an AP
4 $a = 7$
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Sum of First n Terms of an AP
9 $a_{20} - a_{15} = 20$
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nth Term of an AP
9 $14^{\text{th}}$
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nth Term of an AP
11 $3, 6, 9, 12, \ldots, 111$
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nth Term of an AP
12 $Three numbers in A.P. have the sum 30. What is its middle term ?$
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Arithmetic Progressions and Common Difference
15 $The sum of first 200 natural numbers is$
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Sum of First n Terms of an AP
16 $-29, -26, -23, \ldots, 61$
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nth Term of an AP
17 $3n^2 + 4n$
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Sum of First n Terms of an AP
17 $n$
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nth Term of an AP
18 $3p - 1$
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Arithmetic Progressions and Common Difference
26 (a) $If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20 terms.$
Sum of First n Terms of an AP
26 (a) $(m \neq n)$
Sum of First n Terms of an AP
26 (b) $10^{\text{th}}$
Sum of First n Terms of an AP
26 (b) $In an A.P., the sum of three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.$
Arithmetic Progressions and Common Difference
27 (a) $20^{\text{th}}$
Sum of First n Terms of an AP
27 (b) $The first term of an A.P. is 5, the last term is 45 and the sum of all the terms is 400. Find the number of terms and the common difference of the A.P.$
Sum of First n Terms of an AP
31 $A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was ₹ 15,000 after 4 years of service and ₹ 18,000 after 10 years of service, what was his starting salary and what was the annual increment ?$
nth Term of an AP
33 (A) $The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms.$
Sum of First n Terms of an AP
33 (B) $In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the first term and common difference of A.P. Also, find the sum of all the terms of the A.P.$
Sum of First n Terms of an AP
Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize. While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{\text{th}}$ spot is $20 + 4n$ , then answer the following questions to help the players in spotting the clues :
37 (i) $Which number is on first spot ?$
nth Term of an AP
Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize. While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{\text{th}}$ spot is $20 + 4n$ , then answer the following questions to help the players in spotting the clues :
37 (ii) (a) $Which spot is numbered as 112 ?$
nth Term of an AP
Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize. While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{\text{th}}$ spot is $20 + 4n$ , then answer the following questions to help the players in spotting the clues :
37 (ii) (b) $What is the sum of all the numbers on the first 10 spots ?$
Sum of First n Terms of an AP
Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize. While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n^{\text{th}}$ spot is $20 + 4n$ , then answer the following questions to help the players in spotting the clues :
37 (iii) $(n - 2)^{\text{th}}$
nth Term of an AP

2023

30 q

2 $k + 2, 4k - 6$
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Arithmetic Progressions and Common Difference
3 $13^{\text{th}}$
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nth Term of an AP
3 $\sqrt{6}, \sqrt{24}, \sqrt{54}$
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Arithmetic Progressions and Common Difference
3 $n^{\text{th}}$
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nth Term of an AP
3 $p - 1$
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Arithmetic Progressions and Common Difference
7 $\sqrt{7}, \sqrt{28}, \sqrt{63}$
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Arithmetic Progressions and Common Difference
11 $a - 2b - c$
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Arithmetic Progressions and Common Difference
14 $11^{\text{th}}$
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nth Term of an AP
18 $n^{\text{th}}$
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nth Term of an AP
18 $3n^2 + n$
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Sum of First n Terms of an AP
In the question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option out of the following :
20 $2b = a + c$
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Arithmetic Progressions and Common Difference
26 (A) $-14$
nth Term of an AP
26 (B) $Which term of the A.P. : 65, 61, 57, 53, ................. is the first negative term ?$
nth Term of an AP
27 $p^{\text{th}}$
nth Term of an AP
29 (A) $-14$
nth Term of an AP
29 (A) $20^{\text{th}}$
Sum of First n Terms of an AP
29 (B) $30^{\text{th}}$
Sum of First n Terms of an AP
30 $S_n = 6n - n^2$
Sum of First n Terms of an AP
31 $Find the common difference of an A.P. whose first term is 8, the last term is 65 and the sum of all its terms is 730.$
Sum of First n Terms of an AP
32 $45, 39, 33, \ldots$
Sum of First n Terms of an AP
32 (A) $4^{\text{th}}$
Sum of First n Terms of an AP
32 (B) $63q - 3q^2$
Sum of First n Terms of an AP
33 $Prerna saves ₹ 32 during the first month, ₹ 36 in the second month and ₹ 40 in the third month. If she continues to save in this manner, in how many months will she save ₹ 2,000 ?$
Sum of First n Terms of an AP
34 (A) $11^{\text{th}}$
Sum of First n Terms of an AP
34 $1 + 4 + 7 + 10 + \ldots + x = 287$
Sum of First n Terms of an AP
34 (B) $If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.$
Sum of First n Terms of an AP
35 (a) $11^{\text{th}}$
Sum of First n Terms of an AP
35 (A) $Find the sum of integers between 100 and 200 which are (i) divisible by 9 (ii) not divisible by 9.$
Sum of First n Terms of an AP
35 (B) $-4 + (-1) + 2 + 5 + \ldots + x = 437$
Sum of First n Terms of an AP
35 (b) $250 logs are stacked in the following manner : 22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows, are the 250 logs placed and how many logs are there in the top row ?$
Sum of First n Terms of an AP

2022

11 q

1 (a) $-30, -24, -18, \ldots$
Sum of First n Terms of an AP
1 (b) $S_n = n(4n + 1)$
Sum of First n Terms of an AP
2 $How many natural numbers are there between 1 and 1000 which are divisible by 5 but not by 2 ?$
nth Term of an AP
2 (a) $-\frac{11}{2}, -3, -\frac{1}{2}, \ldots$
nth Term of an AP
2 $d = 5$
Sum of First n Terms of an AP
2 (b) $Find a and b so that the numbers a, 7, b, 23 are in A.P.$
Arithmetic Progressions and Common Difference
5 $n^{\text{th}}$
nth Term of an AP
6 $n^{\text{th}}$
Sum of First n Terms of an AP
10 $8^{\text{th}}$
Sum of First n Terms of an AP
In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in different figures. One such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression :
13 (a) $n^{\text{th}}$
nth Term of an AP
In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in different figures. One such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression :
13 (b) $Which figure has 61 matchsticks ?$
nth Term of an AP

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 3 times: 2023*, 2024*, 2026 (* = numbers or wording changed that year)
5 $2x$
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Question 2
Asked 3 times: 2023*, 2024*, 2026 (* = numbers or wording changed that year)
3 $n^{\text{th}}$
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Question 3
Asked 3 times: 2024*, 2025*, 2026 (* = numbers or wording changed that year)
3 $a_{14} - a_8 = 24$
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Question 4
Asked 3 times: 2023*, 2024*, 2026 (* = numbers or wording changed that year)
23 (a) $In an A.P., the first term is 4 and the last term is 31. If sum of all the terms is 175, find the number of terms and the common difference.$
Question 5
Asked 3 times: 2023*, 2024*, 2025 (* = numbers or wording changed that year)
Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the options given below.
20 $10^{\text{th}}$
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Question 6 (case study, 4 parts)
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹ 1,18,000 by paying every month, starting with the first instalment of ₹ 1,000 and he increases the instalment by ₹ 100 every month. Based on the information given above, answer the following questions :
36 (i) $30^{\text{th}}$
36 (ii) $If the total number of instalments is 40, what is the amount paid in the last instalment ?$
36 (iii) (a) $30^{\text{th}}$
36 (iii) (b) $Find the ratio of the tenth instalment to the last instalment.$
Question 7
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
23 (b) $How many terms of the A.P. 21, 18, 15, .... must be added to get the sum zero ?$
Question 8
Asked 2 times: 2022*, 2026 (* = numbers or wording changed that year)
22 (b) $n^{\text{th}}$
Question 9
Asked 2 times: 2024, 2025
3 $Three numbers in AP have the sum 30. What is its middle term ?$
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Question 10
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
33 (a) $The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.$

5.2 Arithmetic Progressions

Solved Examples

Exercise 5.1

5.3 nth Term of an AP

Solved Examples

Exercise 5.2

5.4 Sum of First n Terms of an AP

Solved Examples

Exercise 5.3

Exercise 5.4 (Optional)

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6 (case study, 4 parts)

Question 7

Question 8

Question 9

Question 10