Chapter 5. Arithmetic Progressions– Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 5. Arithmetic Progressions Important Questions with Solutions and Answers

Chapter 5. Arithmetic Progressions

SECTION = A

Question: The 8^th term from the end of the A.P. : 3, 7,11 , 15,………, 143 is [2025 Basic]

(a) 135 (b) 125 (c) 115 (d) 111

Solution: (c) 115

[ Here,

The 8^th term from the end ]

Question: If common difference of an A.P. is – 6 , then value of is : [2023 basic C]

(a) 36 (b) 6 (c) – 36 (d) – 6

Solution: (c) – 36

[ We have, ]

Question: How many terms are there in the A.P. given below ? [2023 Basic ]

14, 19, 24 ,……………., 119

(a) 18 (b) 14 (c) 22 (d) 21

Solution: (c) 22

[ Here,

We have,

]

Question: The seventh term of an A.P. whose first term is 28 and common difference – 4 , is [2022 C]

(a) 0 (b) 4 (c) 52 (d) 56

Solution: (b) 4

[ Here,

We have,

]

Q1. If the numbers , and are in AP, then the value of is

(a) 0 (b) – 2 (c) – 1 (d) 1

Solution : (d) 1

[ We have,

]

Question: In an A.P., if and , then value of is : [2024 Basic]

(a) 3 (b) (c) (d) – 3

Solution: (d) – 3

[ We have,

]

Question: Which of the following is not an A.P. ? [CBSE 2020 (standard)]

(a) (b)

Solution : (c)

[ (a) : ; (b)

Question: The first three terms of an AP respectively are , and ,then equals :

(a) – 3 (b) 4 (c) 5 (d) 2 [2014 Delhi]

Solution : (d) 2

[ Since , and are three consecutive terms of an AP, then

]

Question: The sum of the first three terms of an AP is 30 and the sum of the last three terms is 36. If the first term is 9, then the number of terms is : [2024 Standard]

(a) 10 (b) 5 (c) 6 (d) 13

Solution: (b) 5

[ Here,

The first 3 terms are:

Again, the last three terms of an AP with n terms are:

]

Question: The next term of the A.P : , , , , …….. is :

(a) (b) (c) (d)

Solution : (c)

[ The AP. :

. Here, ,

term ]

Question: term of an AP : is : [SEBA 2019 , 2024 Basic]

(a) 97 (b) 77 (c) – 77 (d) – 87

Solution : (c) – 77

[ Here, , ,

We know that ,

]

Question: The common difference of an A.P. in which is

(a) 7 (b) 8 (c) 5 (d) 6

Solution : (d) 6

[ We have,

]

Question: If and of an A.P. , then is :

(a) 372 (b) 273 (c) 237 (d) 723

Solution : (b) 273

[ We know that ,

]

Question: The common difference of the A.P’s :is : [ CBSE 2013]

(a) (b) (c) (d)

Solution : (d)

[ Common difference ]

Question: The term of an AP’s : 0 , – 4 , – 8 , – 12 , …………….. is : [SEBA 2020]

(a) – 96 (b) – 100 (c) – 104 (d) – 108

Solution : (b) – 100

[ Here , , ,

]

Question: The term of the AP. is : [CBSE 2015 F]

(a) 77 (b) 44 (c) 66 (d) 55

Solution : (d) 55

[ Here ,

]

Fill in the blank

Question: The first three terms of an AP respectively are , and , then equal to .

Solution : 5

[ We have ,

]

Question: The sum of first five multiples of 3 is .

Solution : 45

[ List of numbers becomes : 3 , 6 , 9 , 12 , 15 .

Here ,

]

Question: The next term of the A.P. : , , , , is .

Solution : .

[ The A.P. is : , , , , ;

Here,

term ]

Question: For the AP : , then term is .

Solution : .

[Here,

]

Question: If , and , then is .

Solution : 2

[ We have,

]

Question: The sum of first positive integers .

Solution :

Question: If the common difference of an AP is 5 , then is .

Solution : 20

[ We have , ]

Answer following the question

Question: Find the common difference of an A.P. whose nth term is given by . [2022 Basic]

Solution: Given,

, , , ..............

The common difference

Question: Find a and b so that the numbers a , 7 , b , 23 are in A.P. [2022 Standard]

Solution: We have,

Now,

Again,

Therefore, the value of a and b are – 1 and 15 .

Question: If the term of the A.P. – 1 , 4 , 9 , 14 , ………. is 129 , find the value of . [ CBSE 2017 C]

Solution : Here , ,

A/Q,

Question: Find the value of so that , , in A.P. [ CBSE 2020 (basic)]

Solution : We have ,

Question: Find the term of the A.P. : , ,, , .[CBSE 2020 (basic)]

Solution : Here , , ,

Question: Find the term of the AP : 2 , 7 , 12 , …………

Solution: Here , , ,

We know that ,

Question: In an AP, given , and , then find .

Solution: We know that ,

Question: Write term of an A.P if its term is .

Solution: Given ,

Question: Which term of the AP : 3 , 8 , 13 , 18 , ……………. , is 78 ? [SEBA2015]

Solution : Here , , and

We know that ,

Question: Find the sum of the first 100 positive integers .

Solution : We have ,

Question: Find the sum of an AP’s : 2 , 7 , 12 , …………… , to 10 terms .

Solution : Here , , and

SECTION = B

Question: Find the sum of the first 20 terms of an AP whose nth term is given as. [2022 Standard]

Solution: Given,

, ...............

Here,

We have,

Question: Find the number of terms in the following AP : 5, 11, 17, ....., 203 [2022 Standard]

Solution: Here,

We have,

Therefore, the number of terms is 34 .

Question: In an AP, if a = 50, d = – 4 and , then find the value of n. [2022 Basic]

Solution: Here ,

We have,

(impossible) or

Question: Find the sum of the first twelve 2-digit multiples of 7, using an AP. [2022 Basic]

Solution: List of number multiple of 7 are : 14 , 21 , 28 , ………..

Here,

We have ,

Question: Which term of the AP 2, 11, 20, 29, ..... will be 99 more than its 25th term ? [2022 Basic]

Solution: Here,

A/Q,

Question: Find the number of odd numbers between 20 and 70. [2022 Basic]

Solution: The list of the number of odd numbers between 20 and 70 are: 21 , 23 , 25 , ……… 69 .

Here,

We have,

Question: Find the and terms of an AP : 3 , 8 , 13 , 18 , …………. [SEBA2016]

Solution : Here , and

We know that ,

and

Question: If the term of the A.P : – 1, 4 ,9 ,14, ……….. is 129 ,find the value of . [2017C]

Solution: Here, ; ;

We know that ,

Question: Find the term from the end (towards the first term) of the A.P : 5 , 9 , 13 , ……., 185 . [CBSE 2016]

Solution: We write the given AP in the reverse order then 185 , ……………… , 13 , 9 , 5 .

We know that , ; Here , , ,

Thus , term from the last term is 153 .

Question: In an AP, if , find the AP. [2022 Standard]

Solution: Given ,

We know that ,

Therefore, the AP’s are : 5 , 13 , 21 , ………………

Question: Find the number of terms of the AP : 18 , , 13 , ………………, – 47 .

Solution: Here,

We know that ,

Question: The term of an A.P. is – 4 and its term is – 16 . Find its term .

Solution: Let and be the first term and common difference of an AP respectively .

and

From we get ,

Question: In the following APs, find the missing terms in the boxes :

Solution: Let and be the missing term and also be common difference of an AP.

Given,

A/Q,

Therefore, and

SECTION = C

Question: If the sum of the first 7 terms of an A.P. is – 14 and that of 11 terms is – 55 , then find the sum of its first ‘n’ terms . [2023 Basic]

Solution: Let , a and d be the first term and common difference of an AP respectively .

A/Q,

And

Putting in eq. (i) , we get

Now,

Question: If the sum of the first 6 terms of an AP is 36 and that of the first 16 terms is 256, find the AP. Also, find the sum of its first 10 terms. [2022 Basic]

Solution: Let, and be the first term and common difference of an A.P. respectively.

A/Q ,

and

Putting in equation (i) , we get

The A.P’s are : 1 , 1 + 2 = 3 , 3 + 2 = 5 , 5 + 2 = 7 ,…………

Now ,

Question: 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 . [SEBA 2015]

Solution : Here , , , and let be the common difference .

A/Q,

Again,

[ From ]

From , we get

Therefore, the numbers of terms is 16 and common difference is .

Question: Find the sum of all integers between 50 and 500 which are divisible by 7. [2024 Basic]

Solution: List of all integers between 50 and 500 which are divisible by 7 : 56, 63, 70 ,……………,490 .

Here,

We have,

Question: Determine the AP whose term is 5 and the term is 9 .

Solution: Let and are first term and common difference respectively .

and

From , we get

Hence , the required of an AP : 3 , 4 , 5 , 6 , …………. .

Question: Prerna saves Rs 32 during the first month, Rs 36 in the second month and Rs 40 in the third month. If she continues to save in this manner, in how many months will she save Rs 2,000 ? [2023 Standard]

Solution: Given the A.P’s : 32 , 36 , 40 , ………….

Here,

We have,

(impossible) or

Prerna will save Rs 2,000 in 25 months.

Question: The term of an A.P. is zero . Prove that the term of the A.P. is three times its term . [2016 Delhi]

Solution: Let and are first term and common difference respectively .

A/Q,

and [ From ]

Now, Proved.

Question: Rohan repays his total loan of Rs 1,18,000 by paying every month starting with the first installment of Rs 1,000. If he increases the installment by Rs 100 every month, what amount will be paid by him in the 30th installment ? What amount of loan has he paid after 30th installment ? [2023 Standard ]

Solution: Here,

We have,

So, the 30th instalment is Rs 3900 .

(b) We have,

So, after 30 instalments, he has paid Rs 73500 .

Question: In the following APs, find the missing terms in the boxes .

Solution: Let , , and be the missing term, and also and be first term and common difference of an AP.

and

From , we get

Question: Find how many integers between 200 and 500 are divisible by 8 . [2017 Delhi]

Solution: The list of integers between 200 and 500 are divisible by 8 is : 200 , 208 , 216 , 224 , …………… , 496 .

Here, , ,

We know that,

Question: Which term of the AP : 3 , 15 , 27 , 39 , …………. Will be 132 more than its term ?

Solution : Here , and

A/Q ,

Question: If denotes the sum of first terms of an AP, Prove that .

Solution : let and be the first term and common difference of an AP respectively .

We know that ,

and

[ From ] proved .

Question: Find the term from the last term of the AP : 3 , 8 , 13 , ……………… , 253 .

Solution: We write the given AP in the reverse order : 253 , ……………….. , 13 , 8 , 3

Here , , ,

We know that ,

Question: Find the sum of the AP : .

Solution: Here , ,

and

Now ,

Question: In an AP , given , , find and .

Solution: Given ,

[ ]

Question: Find the sum of the first 22 terms of the AP :

Solution: Here , , ,

Question: Find the sum of the first 15 multiples of 8 .

Solution: The AP’s are : 8 , 16 , 24 , …………………. .

Here, , ,

We have,

Question: Find the sum of first 24 terms of the list of numbers whose term is given by . [ SEBA 2018]

Solution: We have ,

Here ,

Question: How many terms of the AP : 9 , 17 , 25 , …………. must be taken to given a sum of 636 ? [SEBA 2016]

Solution: Here , , ,

We know that,

(impossible)

Therefore, the number of terms is 12 .

Question: If the seventh term of an A.P is and its ninth term is , find its term . [2014 Delhi]

Solution: Let and be the first term and common difference of an AP respectively.

and

From , we get

So,

Question: Determine the AP whose third term is 16 and the term exceeds the term by 12 .

Solution: Let and be the first term and common difference of an AP respectively.

and

From , we get

Question: Find the sum of the first 22 terms of an AP whose common difference is 7 and the term is 149 . [SEBA 2019]

Solution: Here , and

We have ,

and

Question: How many three-digit numbers are divisible by 11 ?

Solution: The list of three digit numbers divisible by 11 is : 110 , 121 , 132 , ……………., 990 .

Here , , and

We know that ,

Question: Find the sum of first 51 terms of an AP whose second and third term are 14 and 18 respectively . [SEBA 2020]

Solution: Let and be the first term and common difference of an AP respectively .

A/Q ,

and

Putting the value of in , we get

Now ,

Question: 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 ?

Solution: The number of rose plants in the rows are : 23 , 21 , 19 , 17 , …………….. , 7 , 5

Let the number of rows in the flower bed be .

Here , , ,

We know that ,

So, there are 10 rows in the flower bed.

Case Study 1 : While buying an expensive item like a house or a car, it becomes easier for a middle-class person to take a loan from a bank and then repay the loan along with interest in easy instalments. Aman buys a car by taking a loan of Rs 2,36,000 from the bank and starts repaying the loan in monthly instalments. He pays Rs 2,000 as the first instalment and then increases the instalment by Rs 500 every month.

(a) Find the amount he pays in the 25th instalment.

(b) Find the total amount paid by him in first 25 instalments. [2022 Standard]

Solution: (a) Here,

We have,

So, the 25th instalment is Rs 14000 .

(b) Here,

We have,

So, the total paid in the first 25 instalments is Rs 200000 .

Case Study 2 : A road roller is a compactor-type engineering vehicle, used to compact soil, gravel, concrete, etc, in the construction of roads and foundations. They are also used at landfills or in agriculture. A company started making road rollers 10 years ago and increased its production uniformly by a fixed number every year. The company produces 800 rollers in the 6^th year and 1130 rollers in the 9^th year.

Based on the above information, answer the following questions : [2024 Standard]

(i) What is the company’s production in the first year ?

(ii) What was the increase in the company’s production every year ?

(iii) (a) What was the company’s production in the 8^th year ?

(b) What was the company’s total production in the first 6 years ?

Solution: Let , a and d be the first term and common difference of an A.P. respectively .

A/Q ,

and

Putting in (i) , we get

i) Production in the first year is 250

(ii) Increase in production every year is 110

(iii) We have,

SECTION = D

Q1. If the term of an A.P. is and term is p , prove that its term is . [CBSE 2017]

Solution: Let and be the first term and common difference of an AP respectively.

and

From , we get

proved .

Question: If the , and terms of an AP are , and respectively ; prove that

Solution: let , and be the first term and common difference of an A.P respectively .

We know that ,

So,

and

proved .

Question: The sum of first 20 terms of an AP is 400 and that of 40 terms is 1600 . Find the sum of first 10 terms and that of terms . [SEBA 2017]

Solution: let and be the first term and common difference of an AP respectively .

A/Q ,

and

Putting the value of in , we get

Now ,

and

Question: If the first term and common difference of an AP are and respectively, then show that .

Solution: Since the first term and common difference of an AP are and respectively .

Again,

and [ From (i) ]

[ From (i) ]

Proved.

Question: The sum of the and term of an AP is 24 and the sum of the and terms is 44 . Find the first three terms of the AP.

Solution: Let the first term and common difference of an AP are and respectively .

According to question,

and

From , we get

Thus , the AP’s are :

i.e. ,

Question: The term of an AP is twice theterm , then show that the term of an AP is twice the term .

Solution: Let and be the first term and common difference of an AP respectively.

According to question,

and

So, Proved.

Question: Find the sum of the integers between 100 and 200 that are : (i) divisible by 9 (ii) not divisible by 9 .

Solution: The list of the integers between 100 and 200 are :

108 , 117 , 126 , ………….., 198 .

(i) Here , , , .

We know that ,

Again,

(ii) We know that , the sum of first 100 positive integers is

And the sum of first 200 positive integers is

Therefore , the sum of the integers between 100 and 200 .

So, the sum of the integers between 100 and 200 that are not divisible by 9

Total number – Total numbers divisible by 9 .

Question: The ratio of the term to the term of an AP is 2 : 3 . Find the ratio of the term to the term and also the ratio of the sum of the first five terms to the sum of the first 21 terms .

Solution : let and be the first term and common difference of an AP respectively .

A/Q ,

and [from ]

Again , [From ]

Question: The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 15 . Find the numbers . [CBSE 2018]

Solution: let , , and are four consecutive number respectively .

A/Q ,

Again,

If and then , , , and

i.e. , – 2 , 2 , 6 and 10 .

If and then , , , and

i.e. , 10 , 6 , 2 and – 2 .

Question: The sum of the first terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first terms of another AP whose first term is – 30 and the common difference is 8 . Find .

Solution: We know that ,

Here , and

Here and

A/Q,

Question: If the term of an A.P is and term is , prove that the sum of first terms of the A.P. is .

Solution: let, and be first term and common difference of an A.P respectively .

We know that ,

and

From , we get

Q12. If , andare in A.P, then prove that , and are also in A.P. [SEBA 2017]

Solution: Since , , and are in A. P.

Again, , and are in A.P.

[ ]

, and are also in A.P.

Question: 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.

Solution: let , and are first term and common difference of an A.P respectively .

A/Q,

and

From and , we get

Putting in (i) Eq., then

Question: If the sum of terms of an A.P. is the same as the sum of its terms, show that the sum of its term is zero.

Solution: let and be first term and common difference of an AP respectively.

A/Q ,

[ From ]

proved .

Posted 5 years ago

Chapter 5. Arithmetic Progressions– Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 5. Arithmetic Progressions Important Questions with Solutions and Answers

Chapter 5. Arithmetic Progressions

Fill in the blank

SECTION = B

SECTION = C

SECTION = D

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