5. Arithmetic Progressions– SEBA Class 10 Mathematics Important Questions and Answers | HSLC Assam Board
Class 10 Mathematics Chapter 5. Arithmetic Progressions– SEBA/HSLC Assam Board Important Questions & Answers
Chapter 5. Arithmetic Progressions
SECTION = A
Question: The 8th 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 8th 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) 
(c) 
(d) 
Solution : (c)
[ (a) : 
; (b) 
(c) : 
; (d) 
]
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 , 
]
SECTION = B
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 
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: 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, 
or 
(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.
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: 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 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
,
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.
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 2 weeks ago