Download JEE Main & Advanced: Sequence and Series PYQs and more Cheat Sheet Mathematics in PDF only on Docsity!
Sequence and Series^1
JEE MAIN PYQ's
Single Correct Type Questions
1. For three positive integers p , q , r , xpq^2 = yqr^ = zp^2 r^ and
r = pq + 1 such that 3, 3 log y x , 3log z y , 7log x z are in A.P.
with common difference 1
. Then r – p – q is equal to
[24 Jan, 2023 (Shift-I)]
(a) 2 ( b ) 6 ( c ) 12 ( d ) –
2. Let s 1 , s 2 , s 3 , ....., s 10 respectively be the sum to 12 terms
of 10 A.P.s whose first terms are 1, 2, 3, ....., 10 and the
common differences are 1, 3, 5, ....., 19 respectively. Then
10
10
s
=
∑ i
i
is equal to [13 April, 2023 (Shift-I)]
( a ) 7380 ( b ) 7220 ( c ) 7360 ( d ) 7260
3. Let S 1 be the sum of first 2 n terms of an arithmetic
progression. Let S 2 be the sum of first 4 n terms of the
same arithmetic progression. If ( S 2 – S 1 ) is 1000, then the
sum of the first 6 n terms of the arithmetic progression is
equal to : [18 March, 2021 (Shift-II)]
( a ) 5000 ( b ) 1000 ( c ) 7000 ( d ) 3000
4. Let Sn denote the sum of the first n -terms of an arithmetic
progression. If S 10^ =^ 530,^ S 5 =^140 , then S^ 20 −^ S 6 is equal
to : [22 July 2021 (Shift-II)]
( a ) 1852 ( b ) 1842
( c ) 1872 ( d ) 1862
5. If 32 sin2α^ – 1, 14 and 34 – 2 sin2α^ are the first three terms of an
A.P. for some α, then the sixth term of this A.P. is:
[5 Sep, 2020 (Shift-I)]
( a ) 65 ( b ) 78 ( c ) 81 ( d ) 66
6. The common difference of the A.P. b 1 , b 2 , ...., bm is 2 more
than the common difference of A.P. a 1 , a 2 , ...., an. If a 40 =
–159, a 100 = –399 and b 100 = a 70 , then b 1 is equal to:
[6 Sep, 2020 (Shift-II)]
( a ) –127 ( b ) –81 ( c ) 127 ( d ) 81
7. The sum of all two digit positive numbers which when
divided by 7 yield 2 or 5 as remainder is:
[10 Jan, 2019 (Shift-I)]
( a ) 1256 ( b ) 1465 ( c ) 1365 ( d ) 1356
8. If 19th^ terms of non-zero A.P. is zero, then its (49th^ term):
(29th^ term) is: [11 Jan, 2019 (Shift-II)]
( a ) 4 : 1 ( b ) 1 : 3 ( c ) 3 : 1 ( d ) 2 : 1
9. If sum of the first 21 terms of the series 1
92 92
log x + log x +
1 94
log x + ..., where x > 0 is 504, then x is equal to:
[20 July 2021 (Shift-II)]
( a ) 7 ( b ) 9 ( c ) 243 ( d ) 81
10. Let a 1 , a 2 , ... a 30 be an A.P.,
30 1
i i
S a
=
= ∑ and
15 2 1 1
i.
i
T a −
=
= ∑ If
a 5 = 27 and S – 2 T = 75, then a 10 is equal to
[9 Jan, 2019 (Shift-I)]
( a ) 52
( b ) 57
( c ) 47
( d ) 42
is equal to (2021)
( a ) 101
( b ) 25
( c ) 101
( d ) 99
Arjuna JEE AIR O1 (2027)
Sequence and Series
AIR JEE MATHEMATICS PREVIOUS YEAR QUESTIONS
2 JEE PYQs Mathematics
12. Let α and b be the roots of x^2 – 3 x + p = 0 and γ and δ be
the roots of x^2 – 6 x + q = 0. If α, b, γ, δ form a geometric
progression. Then ratio (2 q + p ): (2 q – p ) is:
[4 Sep, 2020 (Shift-I)]
( a ) 3 : 1 ( b ) 5 : 3 ( c ) 9 : 7 ( d ) 33 : 31
13. Let a , b , c , d and p be any non zero distinct real
numbers such that ( a^2 + b^2 + c^2 ) p^2 – 2 ( ab + bc + cd ) p +
( b^2 + c^2 + d^2 ) = 0. Then: [6 Sep, 2020 (Shift-I)]
( a ) a , c , p are in G. P. ( b ) a , b , c , d are in A. P.
( c ) a , c , p are in A.P. ( d ) a , b , c , d are in G.P.
14. If α, b and γ are three consecutive terms of a non-constant
G.P. such that the equations α x^2 + 2b x + γ = 0 and x^2 +
x – 1 = 0 have a common root, then α(b + γ) is equal to:
[12 April, 2019 (Shift-II)]
( a ) bγ ( b ) 0 ( c ) αγ ( d ) αb
15. In an increasing geometric series, the sum of the second
and the sixth term is 25
and the product of the third and
fifth term is 25. Then, the sum of 4th, 6th^ and 8th^ terms is
equal to: [26 Feb, 2021 (Shift-I)]
( a ) 30 ( b ) 32 ( c ) 26 ( d ) 35
16. The sum to 10 terms of the series
2 4 2 4 2 4
is
[1 Feb, 2023 (Shift-I)]
( a ) 59
( b ) 55
( c ) 56
( d ) 58
17. Let A 1 and A 2 be two arithmetic means and G 1 , G 2 , G 3 be
three geometric means of two distinct positive numbers.
The G 14 + G 24 + G 34 + G 12 G 32 is equal to
[15 April, 2023 (Shift-I)]
( a ) 2(A 1 + A 2 ) G 1 G 2 ( b ) (A 1 + A 2 )^2 G 1 G 3
( c ) (A 1 + A 2 ) G 12 G 32 ( d ) 2(A 1 + A 2 ) G 12 G 32
18. If Sn = 4 + 11 + 21 + 34 + 50 + ….. to n terms, then
( 29 9 )
S − S [10 April, 2023 (Shift-II)]
( a ) 226 ( b ) 220 ( c ) 223 ( d ) 227
19. If gcd ( m , n ) = 1 and 1^2 – 2^2 + 3^2 – 4^2 + ...... + (2021)^2 –
(2022)^2 + (2023)^2 = 1012 m^2 n , then m^2 – n^2 is equal to
[6 April, 2023 (Shift-II)]
( a ) 200
( b ) 240
( c ) 220
( d ) 180
20. Let x , y > 0. If x^3 y^2 = 2^15 , then the least value of 3 x + 2 y is
[24 June, 2022 (Shift-II)]
( a ) 30 ( b ) 32 ( c ) 36 ( d ) 40
21. If the minimum value of
2 5
f x =^ x + α x >
x
is 14,
then the value of α is equal to [28 July, 2022 (Shift-I)]
( a ) 32 ( b ) 64 ( c ) 128 ( d ) 256
22. If the sum of the first 20 terms of the series
log(7 1/2 ) x + log (71/3 ) x + log (71/4 ) x + ...is 460, then x is equal
to. [5 Sep, 2020 (Shift-II)]
( a ) 72
( b ) e^2
( c ) 7 1/
( d ) 7 46/
Integer Type Questions
23. The 8th^ common term of the series
S 1 = 3 + 7 + 11 + 15 + 19 + ....
S 2 = 1 + 6 + 11 + 16 + 21 + .... is
[30 Jan, 2023 (Shift-II)]
24. The sum of the common terms of the following three
arithmetic progressions. [1 Feb, 2023 (Shift-II)]
2,5,8,11, ………….359 and
2, 7,12,17, ……,197 , is equal to _______.
25. Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59
terms be two series. Then, the sum of the terms common
to both the series is equal to. [29 June, 2022 (Shift-II)]
26. Let a 1 , a 2 ,......, an be in A.P. If a 5 = 2 a 7 and a 11 = 18, then
10 11 11 12 17 18
12 1 1.^1
a a a a a a
+^ +…
+^ +^ + is^ equal^ to
[31 Jan, 2023 (Shift-I)]
27. Let a 1^ ,^ a 2^ ,^ a 3^ ,…^ be a GP of increasing positive numbers.
If the product of fourth and sixth terms is 9 and the sum of
fifth and seventh terms is 24 , then a a 1 9 + a a a 2 4 9 + a 5 + a 7
is equal to [29 Jan, 2023 (Shift-I)]
28. Let 0 < z < y < x be three real numbers such that
x y z
are in an arithmetic progression and x , 2 , y z are in a
geometric progression. If xy + yz
+ zx = xyz , then 3( x
+ y + z )^2 is equal to____ [8 April, 2023 (Shift-II)]
4 JEE PYQs Mathematics
13. Let S 1 , S 2 ,… be squares such that for each n ≥ 1 the length of a side of Sn equals the length of a diagonal of Sn +1. If the length of a side of S 1 is 10 cm , then for which of the following values of n is the area of Sn less than 1 cm^2? [IIT-JEE 1999] ( a ) 7 ( b ) 8 ( c ) 9 ( d ) 10 14. For a positive integer n let
( ) ( )
n − a n , then^ [IIT-JEE 1999]
( a ) a (100) ≤ 100 ( b ) a (100) > 100 ( c ) a (200) ≤ 100 ( d ) a (200) > 100
15. If the first and the (2 n – 1) th term of an AP, GP and HP are equal and their nth terms are a , b and c respectively, then
[IIT-JEE 1988]
( a ) a = b = c ( b ) a ≥ b ≥ c ( c ) a + c = b ( d ) ac – b^2 = 0
Numerical Types/Integer Types Questions
16. Suppose that all the terms of an arithmetic progression are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this AP is
[JEE Adv. 2015]
17. A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k , then k – 20 is equal to
C-23.01 W-19.59 UA-57.4 [JEE Adv. 2013]
18. Let a 1 , a 2 , a 3 , ..., a 100 be an arithmetic progression with a 1 = 3
and 1
P S (^) p = (^) ∑ i = ai ≤. For any integer n with 1 ≤ n ≤ 20, let m
= 5 n. If m n
S
2011] S^ does not depend on^ n , then^ a^2 is equal to ..... [IIT-JEE
19. Let a 1 , a 2 , a 3 , ..., a 11 be real numbers satisfying a 1 = 15, 27 – 2 a 2 > 0 and ak = 2a k–1 – ak– 2 for k = 3, 4, ..., 11.
If
2 2 2 1 2 ...^1190 11
a + a + + a = , then the value of 1 2 ... 11 11
a + a + + a is
... [IIT-JEE 2010]
20. The sum of integers from 1 to 100 that are divisible by 2 or 5 is ...... [IIT-JEE 1984]
21. The interior angles of a polygon are in arithmetic progression.
The smallest angle is 120º and the common difference is 5º. Find the number of sides of the polygon. [IIT-JEE 1980]
22. Let Sk , where k = 1, 2, ..., 100, denotes the sum of the infinite geometric series whose first term is 1 !
k k
− (^) and the common ratio
is 1 k
. Then, the value of (^) ( )
(^2 100 ) 1
100! k^ k
[IIT-JEE 2010]
23. Let m be the minimum possible value of log 3 (3y1^ + 3y2^ +
3 y3), where y 1 , y 2 , y 3 are real numbers for which y 1 + y 2 +
y 3 = 9. Let M be the maximum possible value of (log 3 x 1 +
log 3 x 2 + log 3 x 3 ), where x 1 , x 2 , x 3 are positive real numbers
for which x 1 + x 2 + x 3 = 9. Then the value of log 2 ( m^3 ) + log 3
( M^2 ) is ................
C-28.89W-51.5UA-19.62 [JEE Adv. 2020]
24. Let a 1 , a 2 , a 3 , ..... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b 1 , b 2 , b 3 , .... be a sequence of positive integers in geometric progression with common ratio 2. If a 1 = b 1 = c , then the number of all possible values of c , for which the equality 2( a 1 + a 2 + ... + an ) = b 1 + b 2 + ... + bn holds for some positive integer n , is ......
C-17.16 W-62.2 UA-20.64 [JEE Adv. 2020
25. If cos ( x – y ), cos x and cos ( x + y ) are in HP. Then cos sec 2
x ⋅ ^^ y = … [IIT-JEE^ 1997]
26. If x be is the arithmetic mean and y,z be two geometric means
between any two positive numbers, then
y^3 z^3 xyz
[IIT-JEE 1997]
27. If the harmonic mean and geometric mean of two positive numbers are in the ratio 4 : 5. Then, the two numbers are in the ratio....
[IIT-JEE 1992]
Comprehension/Passage Based Type Questions
Passage-
Let A 1 , G 1 , H 1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An –1 and Hn –1 has arithmetic, geometric and harmonic means as An , Gn , Hn , respectively.
28. Which one of the following statements is correct? [IIT-JEE 2007] ( a ) G 1 > G 2 > G 3 > ... ( b ) G 1 < G 2 < G 3 < ... ( c ) G 1 = G 2 = G 3 = ... ( d ) G 1 < G 3 < G 5 < ... and G 2 > G 4 > G 6 > ... 29. Which of the following statements is correct? [IIT-JEE 2007] ( a ) A 1 > A 2 > A 3 > ... ( b ) A 1 < A 2 < A 3 < ... ( c ) A 1 > A 3 > A 5 > ... and A 2 < A 4 < A 6 < ... ( d ) A 1 < A 3 < A 5 < ... and A 2 > A 4 > A 6 > ...
Sequence and Series^5
30. Which of the following statements is correct? [IIT-JEE 2007]
( a ) H 1 > H 2 > H 3 > ... ( b ) H 1 < H 2 < H 3 < ... ( c ) H 1 > H 3 > H 5 > ... and H 2 < H 4 < H 6 < ... ( d ) H 1 < H 3 < H 5 < ... and H 2 > H 4 > H 6 > ...
Passage-
Let Vr denotes the sum of the first r terms of an arithmetic progression (AP) whose first term is r and the common difference is (2 r – 1). Let Tr = Vr+1 – Vr and Qr = Tr+1 –Tr for r = 1, 2,
31. The sum V 1 + V 2 +..... + V n is [IIT-JEE 2009]
( a ) (^1) ( 1 ) (^) ( 3 2 1 ) 12
n n + n − n + ( b ) (^1) ( 1 ) (^) ( 3 2 2 ) 12
n n + n + n +
( c ) (^1) ( ) (^) ( 2 2 1 ) 2
n n − n + ( d ) (^1 3) ( 2 n^3^ −^2 n +^3 )
- T r is always [IIT-JEE 2009]
( a ) An odd number ( b ) An even number ( c ) A prime number ( d ) A composite number
33. Which one of the following is a correct statement? [IIT-JEE 2009] ( a ) Q 1 , Q 2 , Q 3 , ... are in an AP with common difference 5 ( b ) Q 1 , Q 2 , Q 3 , ... are in an AP with common difference 6 ( c ) Q 1 , Q 2 , Q 3 , ... are in an AP with common difference 11 ( d ) Q 1 = Q 2 = Q 3 = ...
