5. Arithmetic Progressions

Arithmetic Progressions

Chapter 5. Arithmetic Progressions

Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question , 4 Marks and Solutions :

Class 10 Arithmetic Progressions Multiple Questions and Answers

SECTION = A

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,

]

Q2. Which of the following is not an A.P. ? [CBSE 2020 (standard)]

(a)

(b)

(c)

(d)

Solution : (c)

[ (a) : ;

(b)

(d) ]

Q3. The first three terms of an AP respectively are , and ,then equals : [2014 Delhi]

(a) – 3 (b) 4 (c) 5 (d) 2

Solution : (d) 2

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

]

Q4. The next term of the A.P : , , , , …….. is :

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

Solution : (c)

[ The AP. :

Here, ,

term ]

Q5. term of an AP : is : [SEBA 2019]

(a) 97 (b) 77 (c) – 77 (d) – 87

Solution : (c) – 77

[ Here, , ,

We know that ,

]

Q6. The common difference of an A.P. in which is

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

Solution : (d) 6

[ We have,

]

Q7. If and of an A.P. , then is :

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

Solution : (b) 273

[ We know that ,

]

Q8. The common difference of the A.P’s : is : [ CBSE 2013]

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

Solution : (d)

[ Common difference ]

Q9. 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 , , ,

]

Q10. The term of the AP. is : [CBSE 2015 F]

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

Solution : (d) 55

[ Here , , ,

]

Class 10 Arithmetic Progressions Fill in the blank :

Q1. The first three terms of an AP respectively are , and ,then equal to .

Solution : 5

[ We have,

]

Q2. The sum of first five multiples of 3 is .

Solution : 45

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

Here ,

]

Q3. The next term of the A.P. : , , , , is .

Solution : .

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

Here,

term ]

Q4. For the AP : , , , , then term is .

Solution : .

[ Here, , ,

]

Q5. If , and , then is .

Solution : 2

[ We have,

]

Q6. The sum of first positive integers .

Solution : .

Q7. If the common difference of an AP is 5 , then is .

Solution : 20

[ We have ,

]

Class 10 Arithmetic Progressions Answer following the questions

Q1. If the term of the A.P. – 1 , 4 , 9 , 14 , ………. is 129 , find the value of .[ CBSE 2017 C]

Solution : Here , ,

and

Q2. Find the value of so that , , in A.P. [ CBSE 2020 (basic)]

Solution : We have,

Q3. Find the term of the A.P. : , ,, , .[CBSE 2020 (basic)]

Solution : Here , , ,

Q4. Find the term of the AP : 2 , 7 , 12 , …………

Solution: Here , , ,

We know that ,

Q5. In an AP, given , and , then find .

Solution: We know that ,

Q6. Write term of an A.P if its term is .

Solution: Given ,

Q7. Which term of the AP : 3 , 8 , 13 , 18 , ……………. , is 78 ? [SEBA2015]

Solution : Here , , and

We know that ,

Q8. Find the sum of the first 100 positive integers .

Solution : We have ,

Q9. Find the sum of an AP’s : 2 , 7 , 12 , …………… , to 10 terms .

Solution : Here , , and

Case Study Section = II

Case study based questions are compulsory . Attempt any four sub parts of each question .

Each subpart carries 1 marks.

Q1. Reena applied for a job and got selected . She has been offered the job with a starting monthly salary of Rs. 8000 , with an annual increment of Rs. 500 .

Answer the question based upon this situation :

(a) Which of following are A.P ?

(i) 7000 , 7400 , 8400 ,…………

(ii) 8400 , 9600 , 10600 , ………

(iii) 8000 , 8500 , 8900, ………...

(iv) 8500 , 9000, 9500 ,………..

(b) What would be her monthly salary for the fifth year ?

(i) 8500 (ii) 9600 (iii) 10600 (iv) 10000

(i) (ii) (iii) (iv)

(d) The sum of the first 1000 positive integers is:

(i) 50050 (ii) 50500 (iii) 5050 (iv) 500500

Solution: (a) (iv) 8500 , 9000, 9500 ,…………………..

(b) (iv) 10000

[ We have,

]

[ We have,

]

(d) (iv) 500500

[ We have, ]

Q3. In a school, student 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 study, e.g., a section of class will plant 1 tree, a section of class will plant 2 trees and so on till class . There are three sections of each class .

Answer the question based upon this situation :

(a) Which of the following are APs ?

(i) 3 , 5 , 7 , 8 ,…… (ii) 3 , 4 , 7 , 9 ,………

(iii) 3 , 6 , 9 , 11 ,…… (iv) 3 , 6 , 9 , 12 ,…

(b) How many trees planted by class ?

(i) 12 (ii) 24 (iii) 36 (iv) 48

(i) – 31 (ii) – 51 (iii) – 41 (iv) – 21

(d) How many trees will be planted by the students ?

(i) 324 (ii) 423 (iii) 234 (iv) 243

Solution: (a) (iv) 3 , 6 , 9 , 12 ,…………

(b) (iii) 36

[ The number of trees planted by class ]

[ We have , ]

(d) (iii) 234

[ The trees planted by 3 section of class to class are :

3 × 1 , 3 × 2 , 3 × 3 , 3 × 4 , …………….. , 3 × 12

i.e., 3 , 6 , 9 , 12 , …………., 36

Here , , ,

]

Q4: In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line .There are ten potatoes in the line . A competitor starts from the bucket , picks up the nearest potato, runs back with it , drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same away until all the potatoes are in the bucket.

Answer the question based upon this situation :

(a) What is the distance by the competitor to pick up first potato ?

(i) 10 m (ii) 13 m (iii) 16 m (iv) 19 m

(b) What is distance by the competitor to pick up potato ?

(i) 45 m (ii) 43 m (iii) 46 m (iv) 47 m

(i) 370 m (ii) 380 m (iii) 340 m (iv) 350 m

(d) If , , are three consecutive terms of an A.P , then the value of is :

(i) (ii) (iii) (iv) 5

Solution: (a) (i) 10 m

[ The distance by the competitor to pick up first potato m ]

(b) (iii) 46 m

[ The distances (in metres) run by the competitor are : 2×5 ,2 (5 + 3) , 2(5+3+3) ,………

i.e., 10 , 16 , 22 , ……………….

Here , , ,

[ The distances (in metres) run by the competitor are : 2×5 , 2 (5 + 3) , 2(5+3+3) ,………

i.e., 10 , 16 , 22 , ……………….

Here , , ,

]

(d) (iii)

[ We have,

]

Class 10 Arithmetic Progressions 2 Marks Questions and Answers

SECTION = B

Q1. Find the and terms of an AP : 3 , 8 , 13 , 18 , …………. [SEBA2016]

Solution : Here , and

We know that ,

and

Q2. If the term of the A.P : – 1, 4 ,9 ,14, ……….. is 129 ,find the value of . [2017C]

Solution: Here, , ,

We know that,

Q3. 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 .

Q4. In an AP, if , find the AP.

Solution: Given ,

We know that ,

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

Q5. Find the number of terms of the AP : 18 , , 13 , ………………, – 47 .

Solution: Here , ,

We know that,

Q6. 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 ,

Q7. 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,

and

Therefore,

and

Class 10 Arithmetic Progressions 3 Marks Questions and Answers

SECTION = C

Q1. 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 .

Again,

[ From ]

From , we get

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

Q2. 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 , ...... .

Q3. The term of an A.P. is zero . Prove that the term of the A.P. is three times its term . [DELHI 2016]

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

[ From ]

and

Proved.

Q4. 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

Q5. 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,

Q6. Which term of the AP : 3 , 15 , 27 , 39 , …………. Will be 132 more than its term ?

Solution : Here , and

A/Q,

Q7. 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 .

Q8. 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 ,

Q9. Find the sum of the AP : .

Solution: Here , ,

and

Now ,

Q10. In an AP , given , , find and .

Solution: Given ,

Q11. Find the sum of the first 22 terms of the AP :

Solution: Here , , ,

Q12. Find the sum of the first 15 multiples of 8 .

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

Here, , ,

We know that,

Q13. Find the sum of first 24 terms of the list of numbers whose term is given by . [ SEBA 2018]

Solution: We have ,

Here ,

Q14. 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 .

Q15. 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,

Q16. 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

Q17. 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

Q18. 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 ,

Q19. 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,

Q20. 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.

Class 10 Arithmetic Progressions 4 Marks Questions and Answers

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 .

Q2. 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

Again,

proved .

Q3. 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

Q4. 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.

Q5. 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 .

A/Q,

and

From , we get

Thus , the AP’s are :

i.e. ,

Q6. 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.

A/Q,

and

So, Proved.

Q7. 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 .

Q8. 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 ]

Q9. 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 .

Q10. 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,

Q11. If the term of an A.P is and term is , prove that the sum of firstterms 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.

Q13. 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 .

Therefore,

and

From and , we get

Putting in (i) Eq., then

Q14. 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 .