5. Arithmetic Progressions
AP
5. Arithmetic Progressions
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) 
(c) : 
; (d) 
]
Q3. 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
]
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 , 
, 
, 
, 
]
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 , 
]
Answer following the question
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
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
(c) If 
, 
and 
then 
(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, 
]
(c) (iii)
[ 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
(c) If 
form an AP where 
is define
as 
, then the term is :
(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 
]
(c) (iii) – 41
[ 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
way 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
(c) What is the total distance the competitor has to run ?
(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 , 
, 
, 
m
(c) (i) 370 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 , , ,
]
(d) (iii)
[ We have, 
]
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 
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 .
Solution: Let and are first term and common difference respectively . [Delhi16]
[ 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, 
or 
(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.
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, 
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 .
According to question, 
and 
From , we get 
Thus , the AP’s are : 
i.e. , 
Q6. The 
term of an AP is twice the
term , 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.
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 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
,
and
are 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 
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 .
Posted 5 years ago