5. Arithmetic Progressions

SEBA Class 10 Maths Chapter 5. Arithmetic Progressions

Chapter 5. Arithmetic Progressions

Class 10 Maths Chapter 5. Arithmetic Progressions Multiple Questions and Solutions :

Question : 11th term of the AP : is :

(a) 28 (b) 22 (c) – 38 (d)

Solution : (b) 22

[ Here, , ,

We have,

]

Question : The sum of the first 100 positive integer is :

(a) 5050 (b) 5060 (c) 6050 (d) 5005

Solution : (a) 5050

[ Here,

We have,

]

Question: The sum of first 16 terms of the AP : 10 , 6 , 2 , ……… is :

(a) – 320 (b) 320 (c) – 352 (d) – 400

Solution : (a) – 320

[ Here,

We have,

]

Question: The term from the end of the AP : – 11 , – 8 ,– 5 , ……… , 49 is :

(a) 37 (b) 40 (c) 43 (d) 58

Solution: (b) 40

[ Here,

We know that , the term from the end of the AP

The term from the end of the AP ]

Question : Which of the following form an AP ?

(a) 1 , 1 , 2 , 2 , 3 , 3 , ………

(b) 0 , 2 , 0 , 2 , ………..

(d) ……..

Solution: (d) ……..

[ (a) We have ,

(b) We have ,

(d) We have,

Here , , is an AP. ]

Question: 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 : The list of number : – 10 , – 6 , – 2 , 2 , ……… is:

(a) an AP with

(b) an AP with

(d) not an AP

Solution : (b) an AP with

[ Here , First term

Common difference

]

Question : The sum of first 20 odd natural numbers is : [CBSE 2012]

(a) 100 (b) 210 (c) 400 (d) 420

Solution : (c) 400

[ List of odd natural numbers : 1 , 3 , 5 , 7 , ……..

Here,

We have,

]

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

(a)

(b)

(c)

(d)

Solution : (c)

[ (a) : ;

(b)

(d) ]

Question : If term of an A.P. is , then the common difference is :

(a) 7 (b) 10 (c) 12 (d) 6

Solution : (d) 6

[ Given,

The common difference ]

Question: 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

]

Question : The common difference of the AP. : is :

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

Solution : (c)

[ Here, First term

Common difference ]

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]

(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 : Which term of the AP : 21 , 42 , 63 , 84 , ……. Is 210 ?

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

Solution: (b)

[ Here,

We have,

]

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

(a) 32 (b) 20 (c) 24 (d) 30

Solution : (b) 20

[ 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 , , ,

]

Question : If are three consecutive terms of an A.P., then the value of is : [CBSE 2009]

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

Solution : (c)

[ We have,

]

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 : .

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

Class 10 Maths Chapter 5. Arithmetic Progressions 2 Marks Questions and Answers

Q1. Find the and terms of an AP : 3 , 8 , 13 , 18 , …………. [SEBA 2016]

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 .

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 Maths Chapter 5. Arithmetic Progressions 3 Marks Questions and Answers

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 ,

Question : Find the sum of the AP : .

Solution : Here, and

We know that ,

We have,

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

Solution: Given ,

[ Use ]

Question: 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 ,

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 1st , 2nd , 3rd , ………. , 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.

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,

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

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 Maths Chapter 5. Arithmetic Progressions 4 Marks Questions and Answers

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: 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. ,

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: 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 .

Therefore,

and

From and , we get

Putting in (i) Eq., then