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Sequence and Series
Single Correct Type Questions
1. Let, an be the n
th term of an A.P If
100
2 = 1
r^ =
r
a and
100
2 1 1
−^ ,
=
r^ =
r
a then common difference of the A.P.
is:
(A) (^) − (B) −
(C)
2
− (D) None of these
2. If N, the set of natural numbers is partitioned into
groups S 1 ={1}, S 2 = {2, 3}, S 3 = {4,5,6},…, then
the sum digits of the numbers in S 50 is
(A) 20 (B) 10
(C) 18 (D) 19
3. The series of natural numbers is divided into
groups (1), (2,3, 4), (5,6,7,8, 9)... and so on. The
sum of the numbers in the nth group is
(A)
3 3 n + ( n +1) (B)
3 3 ( n −1) + n
(C)
3 3 n + 1 + ( n −1) (D)
3 3 ( n + 1) + ( n −1)
4. The arithmetic mean of two numbers is
and
the positive square root of their product is 15. The
larger of the two numbers is
(A) 24 (B) 25
(C) 20 (D) 30
5. If sec( − 2 ),sec ,sec( + 2 )are in arithmetic
progression then
2 2 cos = cos ( n ; n I )
the value of is:
(A) 1 (B) 2
(C) 3 (D)
6. If S , P and R are the sum, product and sum of the
reciprocals respectively of n terms of an
increasing G.P. and S
n = R
n
. P
k , then k is equal to
(A) 1 (B) 2
(C) 3 (D) None of these
7. If the sum to infinity of the series,
2 3 1 + 4 x + 7 x + 10 x + ….., is
where | x |< 1,
then x equals to:
(A) 19/7 (B) 1/
(C) 1/4 (D) None these
8. If
b + c c + a a + b
are in A.P. then
1 1 1
ax bx cx
x are in
(A) G.P
(B) G. P. only if x < 0
(C) G.P. only if x > 0
(D) None of these
9. If a ii , =1,2,3,4be four positive real numbers,
then the minimum value of
, ,^ {1,2,3,4},^
i
j
a i j i j a
is:
(A) 6 (B) 8
(C) 12 (D) 24
10. If a 1 , a 2 , a 3 , a 4 , a 5 are in H. P, then
a 1 a 2 + a 2 a 3 + a 3 a 4 + a 4 a 5 , is equal to:
(A) 2 a 1 a 5 (B) 3 a 1 a 5
(C) 4 a 1 a 5 (D) – 4
11. If a 1 , a 2 , a 3 , …, an are in HP and
1
=
n
r K r
f K a a
Then
1 2 , , , (1) (2) ( )
a a an
f f f n
Are in
(A) AP (B) GP
(C) HP (D) AGP
12. If the roots of the equation 10 x
3
2
=0 are in HP, then K is equal to
(A) 3 (B) 6
(C) 9 (D) 12
Arjuna JEE AIR O 1 (2027)
Practice Sheet [Legend]
13. If a, b, c are in H.P., then the value of
3 3 3 3 3 3
2 2
a b + b c + c a
a c
is
(A)
2 9 ac − 6 b (B)
2 3 ac − 2 b
(C)
2 9 ac − 4 b (D)
2 9 ac − 2 b
14. For the series 21, 22, 23,…, K– 1, K ; the A. M. and
G .M. of the first and last numbers exist in the
given series. If ‘ k ’ is a three digits number, then
‘ k ’ can attain
(A) 5 values (B) 6 values
(C) 2 values (D) 4 values
15. If x > 0, then the minimum value of
x + x + x + x + x
is
(A) 1000 (B) 2000
(C) 1996 (D) 3000
( )
( )( )
2 3 2
4 2 2 1
=
r
r r
r r r r
is equal to
(A)
(B) 1
(C) 2 (D) infimite
17. The sum of the series
(^2 5 1 10 2 ) 2 2 2 .. 1.2 2.3 3.4 4.
equal:
(A)
n n
n
(B) 2 1
n n
n
(C)
n n
n
(D)
n n
n
One or More Than One Correct Type Questions
18. It is given that the sequence a n satisfies
a 1 (^) = 0, an (^) + 1 = an + 1 + 2 1+ an for n N .Then
(A) a 100 = 9999 (B) a 2001 = 4004000
(C) a 2001 = 4002000 (D) a 19 = 360
19. The sequence an^ ^ ,^ n^ N satisfies a 1 = 1 and
5 an^ +^^1 − an^1
2
n +
then
(where [ ] denotes greatest integer function)
(A) (^) a 501 = 3 (B) a 207 = 3
(C) (^) a 223 = 4 (D) a 625 = 4
20. Let (^) a 1 (^) , a 2 (^) , a 3 , , an be the first ‘ n ’ terms of an A.P
having common difference ‘d’( d 0), then the
greatest value of product of two terms equidistant
from the extreme terms is:
(A)
2 2
1
n +
d n (^) a a if n is odd
(B)
2 2
1
n +
d n a a if n is odd
(C)
2
1
n +
d n n (^) a a if n is even
(D)
2
1 (^ 2) 4
n^ +^ −
d a a n n is n is even
21. If
a b c
are in AP and a , b, – 2 c are in GP, where
a, b, c are non - zero, then
(A)
3 3 3 a + b + c = 3 abc
(B) (^) − 2 a , b , − 2 c are in AP
(C) −2 , ,^ a b^^ −^2 c are in GP
(D)
2 2 2
a , b , 4 c are in GP
22. For all permissible values of x , consider
sin3 (cos6 cos 4 )
sin (cos8 cos 2 )
x x x y x x x
and range of y is
( −, ) a ( , b ).^ If 2 b^ is the first term of G. P and
‘ a ’ is its common ratio, then ( S ∞ denotes the some
of
infinite terms of G.P)
(A)
b − a =
(B) 3 a + b = 4
(C) s = 9
(D)
s (^) = a + b
23. If a , b , c are 3 distinct numbers in H.P .,
a,b,c > 0, then:
(A) , ,
b + c − a c + a − b a + b − c
a b c
are in A.P
(B) , ,
b + c c + a a + b
a b c
are in A. P
(C)
5 5 5 a + c 2 b
(D)
a b a
b c c
36. The sum
1
−
= =
n
n k n k
k is equal to
(A)
=
(^) k k
k
(B)
=
(^) k k
k
(C)
1 1
= = +
(^) m n m n m
m
(D)
Paragraph Type Questions
Passage-I
There are two sets A and B each of which consists of
three numbers in A.P. whose sum is 15. D and d are their
respective common differences such that
D − d = 1, D 0. If
p
q
where p and q are the product
of the numbers in those sets A and B respectively.
37. Sum of the product of the numbers in set A taken
two at a time is:
(A) 51 (B) 71
(C) 74 (D) 86
38. Sum of the product of the numbers in set B taken
two at time is:
(A) 52 (B) 54
(C) 64 (D) 74
Passage-II
The first four terms of a sequence are given by T 1 =0,
T 2 =1, T 3 =1, T 4 =2. The general term is given by
− 1 − 1 = +
n n Tn A B where A, B , , are independent of
n and A is positive.
39. The value of( )
2 2 + + is equal to:
(A) 1
(B) 2
(C) 5
(D) 4
40. The value of ( )
2 2 5 A + B is equal to:
(A) 2 (B) 4
(C) 6 (D) 8
Passage-III
Let x, y, z are positive real numbers and x + y + z = 60
and x >3.
41. Maximum value of (^) ( x − 3)( y + 1)( z +5)is:
(A) (17) (21) (25) (B) (20) (21) (23)
(C) (21) (21) (21) (D) (23) (19) (15)
42. Maximum value of ( x – 3) (2 y +1) (3 z +5) is :
(A)
3
3 2
(B)
3
3 3
(C)
3
2 3
(D)
3
2 2
43. Maximum value of xyz is:
(A)
3 8 10
(B)
3 27 10
(C)
3 64 10
(D)
3 125 10
Passage-IV
If
r = + + + + + r
and
1
=
^ +^ ^ =^ ^ +^ −
n
r
r r P n n Q n where P ( n ) and Q
( n )are polynomial function of ‘ n ’, then
10
0
=
r
P r is equal to
(A) 235 (B) 506
(C) 285 (D) 385
0
=
r Q r
is equal to
(A) 1
(B) 2
(C) 4
(D) 8
46. P (13) − Q (13) is equal to
(A) 81 (B) 78
(C) 91 (D) 68
Matrix Match Type Questions
47. Match the list and choose the correct option
List–I List–II
I Suppose that
F n F n for n
= 1, 2, 3,… and F (1) = 2.
Then F (101) equals
P 42
II If a 1 , a 2 , a 3 , …… a 21 are in
A.P. and a 3 + a 5 + a 11 + a 17 + a 19
= 10 then the value of
21
= 1
i i
a
is
Q 1620
III (^) 10 th^ term of the sequences S =
1 + 5 + 13 + 29 + …… is
R 52
IV The sum of all two digit
numbers which are not
divisible by 2 or 3 is
S 2045
T 2 + 4 + 6
+….+ 12
I II III IV
(A) R P, T S Q
(B) T P, R Q S
(C) R P, S Q R
(D) T P,T S Q
48. Match the list and choose the correct option
List–I List–II
I The arithmetic mean of two
positive numbers is 6 and their
geometric mean G and harmonic
mean H satisfy the relation G^2 +
3H = 48 , then product of the
two number is.
P^2
7
II (^) The sum of the series 5 12 ⋅ 42
11
42 ⋅ 72
17
72 ⋅ 102
Q 32
III If the first two terms of a
Harmonic Progression be
1
2
and
1
3
,
then the Harmonic Mean of the
first four terms is
R^1
3
IV Geometric mean of 4 and 9 S 6
T – 6
I II III IV
(A) Q R, T P, S T
(B) S R, P T Q
(C) S R, T P, S T
(D) Q R P S
49. Consider a sequence b n of integers such that
b b 1 , 2 (^) , b 3 are^ in^ G.P.^ b 2 (^) , b 3 (^) , b 4 are^ in^ A.P.,
b 3 (^) , b 4 (^) , b 5 are in G.P., b 4 (^) , b 5 (^) , b 6 are in A.P.,
b 5 (^) , b 6 (^) , b 7 are in G.P. and so on. Also given that
b 1 = 1 and b 5 (^) + b 6 = 198. Then match list below
and choose the correct option
List–I List–II
I b 7 is equal to
P 5
II Sum of digits of b 8 is equal to Q^ 15
III b 9 is equal to
R 9
IV Sum of digits of b 10 is equal
to
S
17
T 13
I II III IV
(A) T P S Q
(B) P S Q T
(C) T R P S
(D) T Q R P
50. Match the list and choose the correct option
List–I List–II
I Let a , b , c are positive real
numbers such that
3 2 a b c = 12 ,
then the minimum value of
49 a + 3 b + c^ is equal to
P 1
II The minimum value of
3 2
2 x − x
for x < 0 equal to
Q 5
III The maximum value of
( )
5 3 8
x − x
for 0 < x < 2 is
equal to
R 7
IV (^) If x^7 y^5 = a and 7 x + 5 y ≥ 12 x ,
y > 0, then the minimum value
of ‘ a ’ is equal to
S 15
T 42
I II III IV
(A) T Q S Q
(B) T Q, S R P, Q
(C) Q R, S S, Q P, Q
(D) T Q S P
ANSWER KEY
1. (D)
2. (A)
3. (B)
4. (D)
5. (B)
6. (B)
7. (B)
8. (A)
9. (C)
10. (C)
11. (C)
12. (C)
13. (A)
14. (C)
15. (B)
16. (A)
17. (A)
18. (A, B, D)
19. (A, B, D)
20. (A, D)
21. (A, B, D)
22. (B, C, D)
23. (A, B, C, D)
24. (B, D)
25. (A, B, C, D)
26. (A, B, C)
27. (A, B, C)
28. (B, C)
29. (A, B, C)
30. (B, C)
31. (A, B, D)
32. (C, D)
33. (A, C)
34. (B, C)
35. (A, B, C, D)
36. (B, C)
37. (B)
38. (D)
39. (B)
40. (A)
41. (C)
42. (A)
43. (A)
44. (D)
45. (B)
46. (C)
47. (A)
48. (D)
49. (A)
50. (D)
51. (C)
Hints and Solution
1. (D)
= 100 a + (1 + 3 + + 5 199) d
= 100 a + (0 + 2 ++198) d
− = d =
2. (A)
1 50 1
of (1 2 49) 50 1 2
T S +
S 50 = {1226, 1227, ,1275}
Sum of number of 50
S = + =
3. (B)
T 1 of nth division = (1 + 3 + 5 + … + 2 n – 3) + 1
= ( n – 1)
2
Number of terms 1 + ( n + 1)2 = 2 n – 1
4. (D)
and 15 2 4
a b ab
a
x x
b
x x
x =
5. (B)
2sec = sec( + +2 ) sec( − 2 )
2 cos( 2 ) cos( 2 )
cos cos( 2 )cos( 2 )
2 2 2 cos − sin 2 = cos cos 2
2
2
cos 2 cos
6. (B)
( )
( 1)
2 1
n (^) n n n n n
a r (^) r S P a r R r (^) a r r
−
n Put in = gives =2.
n k S P R k
7. (B)
2 3 S = 1 + 4 x + 7 x + 10 x +
2 3 xS = x + 4 x + 7 x +
2 3 (1 − x S ) = 1 + 3 x + 3 x + 3 x +
x S x x x
−^ −
8. (A)
c a c b a b
on simplifying gives 2 b = a + c
a b , , c are in AP
ax + 1, bx + 1, cx + 1 → AP
1 1 1 9 ,9 ,.
ax bx cx GP
→
9. (C)
x 2 if x 0 x
(^) 12
i
j
a
a
10. (C)
Using
1 3 3 5 1 5 2 4 3 1 3 3 5 1 5
a a a a a a a a a a a a a a a
gives a a 1 2 (^) + a a 2 3 (^) + a a 3 4 (^) + a a 4 5 (^) = 4 a a 1 5
11. (C)
1 2 3
n
A P
a a a a
1 2
n
a a a AP a a a
1 2
1 2
n
n
f a f a f n a AP a a a
1 2
n
f f f n AP a a a
1 2 , , , (1) (2) ( )
a a a n H P f f f n
12. (C)
Let roots , and
Back substituting in the equation gives k =
13. (A)
3 3 3 3 3 2 ( ab + bc ) = a b + b c + 3 ab c ab ( + bc )
3 3 3 2 3 3 3 3 2 2 2
2 2 2 2
a b b c c a 9 a c 6 a b c
a c a c
2 = 9 ac − 6 b
14. (C)
k AM GM k
11
−
n
n n
25. (A, B, C, D)
Since a 912 , a 951 and a 480 , is divisible by 3
Now,
91 91 7
(^91 )
a
( )( )
7 84 8 = 1 + 10 + +. 10 1 + 10 + +.. 10
a 91 is not prime
26. (A, B, C)
1/ 1 3 5 (2^ 1)
(a) (1.3.5 (2 1))
n n n n n
1 (1.3.5 (2 n − 1)) n
( )
2 1 1 1/ 1 2 2 .. (^2) 1 2 ( 1) (^2) (b) 2 2
n n n n
n
− −
1
2 1 22
n n n
−
+
(c) .. n 1 n 1 n 1 2 n
2 n 2 n 2
27. (A, B, C)
We have,
1 1 (^ 1)^ 1
n r
a a a a r n
1 (^ )^ (^ 1)
a n r a n r
n
1 1 1 1
n n r n n
a a n h a n n a a r a
− +
1 (^1 )
1
n n
n
a a n a a
a r a n r
we get a hr n r (^) − + 1 = a a 1 n = an r (^) − + 1 hr
28. (B, C)
2 D 1 (^) : b − 4 ac 0
2 D 2 (^) : c − 4 ab 0
2 D 3 (^) : a − 4 bc 0
2 2 2 D 1 (^) + D 2 (^) + D 3 (^) : a + b + c 4( ab + bc + ac )
2 2 2
1 4
a b c
ab bc ac
29. (A, B, C)
2 2 2 1 1 1 2
a b c a b c
b c c a a b b c c a a b
a b c b c c a a b
^ +^ +^ + ^ ^
30. (B, C)
a c ce b d c e
2 2 If c = bd , then c = 36
( a = 2, e =18)
31. (A, B, D)
2 2 2 2 1 1
n n
n n r r r
r S S t
= = r
1
n
r
r r n
+^ +
32. (C, D)
2 4 2
n
x x x n x x x
4 4 8
x (^1) x 1 x 1 x 1
1
2 2 1
n
n n
x n x x
f x x x
33. (A, C)
2 terms 4 terms
S n
1
1
sterms (^2) terns
n
n n n
−
−
+^ + + +^ +^ +^
1 1 1
n − n − n −
34. (B, C)
S = + + +
s + + + +
35. (A, B, C, D)
S = 3(1 + 2 + 3 + 4 + ++2016)
4 3 = 2 3 7 2017
2 2 2 2 2 2 (^6) + 5 + 4 − 3 − 2 − 1
= 3[(6 + 3) + (5 + 2) + (4 +1)
= 3(1 + 2 + 3 + 4 + 5 +6)]
36. (B, C)
1
1 1 1 1
n
n k k n n k k n k
k k
−
= = = = +
1 1
k k k
k
=
= (^)
1
4 k^9 k
k
=
^ =
37. (B), 38 (D)
Set A : 5 − D ,5,5 + D and
SetB : S − d ,5,5+ d
2
2
p D
q (^) d
2 2 2 25 = 8 D − 7 d = d + 16 d + 8
d = 1 and D = 2
( D = 1 + d )Set A{3, 5, 7}and B {4,5,6}
39. (B), 40. (A)
T 1 = A + B = 0
A = − B
T 2 + A + B = 1 A ( − =) 1
2 2 T 3 (^) + A + B = (^1) ( )
2 2 A − = 1
3 3 T 4 (^) + A + B = (^2) ( )
3 3 A − = 2
+ = 1 and = − 1
41. (C)
[( 3)( 1)( 5)]
x y z x y z
42. (A)
Term is 6( 3) 2 3
x y z
1/
x y z
x y z
^ ^
43. (A)
1/3 3 ( ) ; (20) 3
x y z xyz xyz
44. (D), 45. (B) 46. (C)
( )
2 2
1
n
r
r r r
=
^ +^ −^
2
1
( 1) ( ( ) ( 1))
n
r
r r r
=
= (^) + − + +
( )
2 2
1
n
r
r r r r
=
^ +^ ^ +^ −^
2 2
1
n
r
r n n
=
= − (^) + + + + −
2 = − (1 + 2 + 3 + +. ( n + 1)) + ( n + 1) ( n +1)
n n n n
47. (A)
(i) F (1), F (2), F (3), is an AP with common
difference
(ii) a 1 (^) + 2 d + a 1 (^) + 4 d + a 1 + 10 d +
a 1 (^) + 16 d + a 1 (^) + 18 d = 5 a 1 + 50 d
a 1 (^) + 2 d + a 1 (^) + 4 d + a 1 (^) + 10 d + a 1 +
16 d + a 1 + 10 d = 2
= (^5) ( a 1 (^) + 10 d (^) )= 10 i.e. a 1 + 10 d = 2
( )
21
1 1 1 1
Now, 2 20 21 10 42 2 i
a a d a d
=
^ =^ +^ =^ +^ =
(iii) S = 1 + 5 + 13 + 29 + +.. t 10
S = 1 + 5 + 13 + +.. t 8 (^) + t 10
Subtrating t 10 = 1 + 4 + 8 + 16 +^ upto^10
terms
(iv) Sum of all two-digit numbers
90 (10 99) (45)(109) 2
Sum of all two-digit numbers is divisible by
Sum of all two-digit numbers is divisible by
30 3 (12 99) 15(111) 2
2 k 9 k + ( x - 4) k + 4 = 0
2 D 0 ( x − 4) − 144 0
x − −( , 8] [16, )
r s r s
1 and 7
r r s r s s r
3 2 7 r − 6 r + 21 r − 18
( )^ (^ )
2 r + 3 7 r − 6 = 0
and 7 14
r = s =
Adding, x y z 2018
x + y + z = 2018 3
x y z
x y z
x = y = z = 2018
( )
2018 2 2018
1 1
= =
r r r r^ r r
x x x x
( )
6
1
i i i
a b a b
=
( )
6 6 (^2 )
1 1
i i i^6 i i
a a a b a
= =
−^ +^ =
2 − 2 a (^) a (^) i + (^) ai + a bi i
1
8 𝑛^4
𝑛 [𝑘(𝑘 + 2 )(𝑘 + 4 )(𝑘 + 6 )
−(𝑘 − 2 )(𝑘 + 2 )(𝑘 + 4 )]
4
1 ( 1)( 1)( 3)( 5) ( 2)( 4)( 6) 15
8
n n n n n n n n
n
− + + + + + + + +
If a x y z b , , , ,
A.P. , and 4 2 4
a b a b z b x y z
If a x y z b , , , ,
H.P. , and 3 3
ab ab ab x y z b a a b a b
If 55 4 2 2
^ a^ +^ b^ a^ +^ b^ a^ + b =
and 7 3 3 55
ab ab ab ab b a a b a b
=^ ^ =
+^ +^ +
0 0 1 2 3 2 2 2 1{2}
− −
− − −
− − −
2 3 4 1 2 2 2 3 2 4 2 .. 2
n S = + + + + + n
2 3 4 2 S = 1 2 + 2 2 + 3 2 +.
1 ( 1) 2 2
n n n n
1 10 ( 1) 2 2 2 2
n n S n
− + = − + = +
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