Sequences and Series

372 Questions
2021 JEE Mains MCQ
JEE Main 2021 (Online) 31st August Morning Shift
Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r2, then r2 $-$ d is equal to :
A.
7 $-$ 7$\sqrt 3 $
B.
7 + $\sqrt 3 $
C.
7 $-$ $\sqrt 3 $
D.
7 + 3$\sqrt 3 $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Evening Shift
If 0 < x < 1 and $y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ....$, then the value of e1 + y at $x = {1 \over 2}$ is :
A.
${1 \over 2}{e^2}$
B.
2e
C.
${1 \over 2}\sqrt e $
D.
2e2
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Morning Shift
If 0 < x < 1, then ${3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + .....$, is equal to :
A.
$x\left( {{{1 + x} \over {1 - x}}} \right) + {\log _e}(1 - x)$
B.
$x\left( {{{1 - x} \over {1 + x}}} \right) + {\log _e}(1 - x)$
C.
${{1 - x} \over {1 + x}} + {\log _e}(1 - x)$
D.
${{1 + x} \over {1 - x}} + {\log _e}(1 - x)$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Morning Shift
If for x, y $\in$ R, x > 0, y = log10x + log10x1/3 + log10x1/9 + ...... upto $\infty$ terms

and ${{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}$, then the ordered pair (x, y) is equal to :
A.
(106, 6)
B.
(104, 6)
C.
(102, 3)
D.
(106, 9)
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th August Morning Shift
The sum of the series

${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$ when x = 2 is :
A.
$1 + {{{2^{101}}} \over {{4^{101}} - 1}}$
B.
$1 + {{{2^{100}}} \over {{4^{101}} - 1}}$
C.
$1 - {{{2^{100}}} \over {{4^{100}} - 1}}$
D.
$1 - {{{2^{101}}} \over {{2^{400}} - 1}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th August Morning Shift
If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ....... is :
A.
${5 \over 2}$
B.
${1 \over 2}$
C.
${25 \over 2}$
D.
${9 \over 2}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th July Morning Shift
Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :
A.
6
B.
4
C.
2
D.
8
2021 JEE Mains MCQ
JEE Main 2021 (Online) 22th July Evening Shift
Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 $-$ S6 is equal to:
A.
1862
B.
1842
C.
1852
D.
1872
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Evening Shift
If sum of the first 21 terms of the series ${\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .......$, where x > 0 is 504, then x is equal to
A.
243
B.
9
C.
7
D.
81
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 $-$ S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :
A.
7000
B.
1000
C.
3000
D.
5000
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Morning Shift
If $\alpha$, $\beta$ are natural numbers such that
100$\alpha$ $-$ 199$\beta$ = (100)(100) + (99)(101) + (98)(102) + ...... + (1)(199), then the slope of the line passing through ($\alpha$, $\beta$) and origin is :
A.
540
B.
550
C.
530
D.
510
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Morning Shift
${1 \over {{3^2} - 1}} + {1 \over {{5^2} - 1}} + {1 \over {{7^2} - 1}} + .... + {1 \over {{{(201)}^2} - 1}}$ is equal to
A.
${{101} \over {404}}$
B.
${{25} \over {101}}$
C.
${{101} \over {408}}$
D.
${{99} \over {400}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
The sum of the series

$\sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}} $ is equal to :
A.
${{41} \over 8}e + {{19} \over 8}{e^{ - 1}} - 10$
B.
${{41} \over 8}e - {{19} \over 8}{e^{ - 1}} - 10$
C.
${{41} \over 8}e + {{19} \over 8}{e^{ - 1}} + 10$
D.
$ - {{41} \over 8}e + {{19} \over 8}{e^{ - 1}} - 10$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
The sum of the infinite series
$1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ......$ is equal to :
A.
${9 \over 4}$
B.
${13 \over 4}$
C.
${15 \over 4}$
D.
${11 \over 4}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
In an increasing geometric series, the sum of the second and the sixth term is ${{25} \over 2}$ and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :
A.
30
B.
32
C.
26
D.
35
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Evening Shift
The minimum value of $f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$, where a, $x \in R$ and a > 0, is equal to :
A.
$a + {1 \over a}$
B.
2a
C.
a + 1
D.
$2\sqrt a $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Morning Shift
If $0 < \theta ,\phi < {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi } $ and $z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta .{{\sin }^{2n}}\phi } $ then :
A.
xy $-$ z = (x + y)z
B.
xyz = 4
C.
xy + z = (x + y)z
D.
xy + yz + zx = z
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Evening Slot
The common difference of the A.P.
b1, b2, … , bm is 2 more than the common
difference of A.P. a1, a2, …, an. If
a40 = –159, a100 = –399 and b100 = a70, then b1 is equal to :
A.
127
B.
81
C.
–127
D.
-81
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Morning Slot
Let a , b, c , d and p be any non zero distinct real numbers such that
(a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then :
A.
a, c, p are in G.P.
B.
a, b, c, d are in G.P.
C.
a, b, c, d are in A.P.
D.
a, c, p are in A.P.
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Evening Slot
If the sum of the first 20 terms of the series
${\log _{\left( {{7^{1/2}}} \right)}}x + {\log _{\left( {{7^{1/3}}} \right)}}x + {\log _{\left( {{7^{1/4}}} \right)}}x + ...$ is 460,
then x is equal to :
A.
e2
B.
71/2
C.
72
D.
746/21
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Evening Slot
If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is :
A.
${2 \over {13}}\left( {{3^{50}} - 1} \right)$
B.
${1 \over {13}}\left( {{3^{50}} - 1} \right)$
C.
${1 \over {26}}\left( {{3^{49}} - 1} \right)$
D.
${1 \over {26}}\left( {{3^{50}} - 1} \right)$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Morning Slot
If ${3^{2\sin 2\alpha - 1}}$, 14 and ${3^{4 - 2\sin 2\alpha }}$ are the first three terms of an A.P. for some $\alpha $, then the sixth terms of this A.P. is:
A.
66
B.
81
C.
65
D.
78
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Morning Slot
If 210 + 29.31 + 28 .32 +.....+ 2.39 + 310 = S - 211, then S is equal to :
A.
${{{3^{11}}} \over 2} + {2^{10}}$
B.
311 — 212
C.
2.311
D.
311
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
Let a1, a2, ..., an be a given A.P. whose
common difference is an integer and
Sn = a1 + a2 + .... + an. If a1 = 1, an = 300 and 15 $ \le $ n $ \le $ 50, then
the ordered pair (Sn-4, an–4) is equal to:
A.
(2480, 249)
B.
(2480, 248)
C.
(2490, 248)
D.
(2490, 249)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
The minimum value of 2sinx + 2cosx is :
A.
${2^{-1 + \sqrt 2 }}$
B.
${2^{1 - {1 \over {\sqrt 2 }}}}$
C.
${2^{1 - \sqrt 2 }}$
D.
${2^{-1 + {1 \over {\sqrt 2 }}}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Morning Slot
If 1+(1–22.1)+(1–42.3)+(1-62.5)+......+(1-202.19)= $\alpha $ - 220$\beta $,
then an ordered pair $\left( {\alpha ,\beta } \right)$ is equal to:
A.
(11, 103)
B.
(10, 103)
C.
(10, 97)
D.
(11, 97)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
If the sum of the series

20 + 19${3 \over 5}$ + 19${1 \over 5}$ + 18${4 \over 5}$ + ...

upto nth term is 488 and the nth term is negative, then :
A.
n = 41
B.
n = 60
C.
nth term is –4
D.
nth term is -4${2 \over 5}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :
A.
${1 \over 4}$
B.
${1 \over 5}$
C.
${1 \over 7}$
D.
${1 \over 6}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
Let S be the sum of the first 9 terms of the series :
{x + k$a$} + {x2 + (k + 2)$a$} + {x3 + (k + 4)$a$}
+ {x4 + (k + 6)$a$} + .... where a $ \ne $ 0 and x $ \ne $ 1.

If S = ${{{x^{10}} - x + 45a\left( {x - 1} \right)} \over {x - 1}}$, then k is equal to :
A.
-3
B.
1
C.
-5
D.
3
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
If the sum of first 11 terms of an A.P.,
a1, a2, a3, .... is 0 (a $ \ne $ 0), then the sum of the A.P.,
a1 , a3 , a5 ,....., a23 is ka1 , where k is equal to :
A.
${{121} \over {10}}$
B.
-${{121} \over {10}}$
C.
${{72} \over 5}$
D.
-${{72} \over 5}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :
A.
[-3, $\infty $)
B.
(-$ \propto $, 9]
C.
(-$ \propto $, -9] $ \cup $ [-3, $\infty $)
D.
(-$ \propto $, -3] $ \cup $ [9, $\infty $)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
If |x| < 1, |y| < 1 and x $ \ne $ y, then the sum to infinity of the following series

(x + y) + (x2+xy+y2) + (x3+x2y + xy2+y3) + ....
A.
${{x + y - xy} \over {\left( {1 + x} \right)\left( {1 + y} \right)}}$
B.
${{x + y - xy} \over {\left( {1 - x} \right)\left( {1 - y} \right)}}$
C.
${{x + y + xy} \over {\left( {1 + x} \right)\left( {1 + y} \right)}}$
D.
${{x + y + xy} \over {\left( {1 - x} \right)\left( {1 - y} \right)}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Evening Slot
Let an be the nth term of a G.P. of positive terms.

$\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200} $ and $\sum\limits_{n = 1}^{100} {{a_{2n}} = 100} $,

then $\sum\limits_{n = 1}^{200} {{a_n}} $ is equal to :
A.
150
B.
175
C.
225
D.
300
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
The product ${2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}}$ ... to $\infty $ is equal to :
A.
${2^{{1 \over 4}}}$
B.
${2^{{1 \over 2}}}$
C.
1
D.
2
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Evening Slot
If the 10th term of an A.P. is ${1 \over {20}}$ and its 20th term is ${1 \over {10}}$, then the sum of its first 200 terms is
A.
100
B.
$100{1 \over 2}$
C.
$50{1 \over 4}$
D.
50
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Morning Slot
Let ƒ : R $ \to $ R be such that for all x $ \in $ R
(21+x + 21–x), ƒ(x) and (3x + 3–x) are in A.P.,
then the minimum value of ƒ(x) is
A.
2
B.
0
C.
3
D.
4
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Evening Slot
If the sum of the first 40 terms of the series,
3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :
A.
20
B.
5
C.
10
D.
25
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Evening Slot
Let ${a_1}$ , ${a_2}$ , ${a_3}$ ,....... be a G.P. such that
${a_1}$ < 0, ${a_1}$ + ${a_2}$ = 4 and ${a_3}$ + ${a_4}$ = 16.
If $\sum\limits_{i = 1}^9 {{a_i}} = 4\lambda $, then $\lambda $ is equal to:
A.
171
B.
-171
C.
-513
D.
${{511} \over 3}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Morning Slot
Five numbers are in A.P. whose sum is 25 and product is 2520. If one of these five numbers is -${1 \over 2}$ , then the greatest number amongst them is:
A.
${{21} \over 2}$
B.
27
C.
7
D.
16
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
If a1, a2, a3, ..... are in A.P. such that a1 + a7 + a16 = 40, then the sum of the first 15 terms of this A.P. is :
A.
120
B.
200
C.
150
D.
280
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
For x $\varepsilon $ R, let [x] denote the greatest integer $ \le $ x, then the sum of the series $\left[ { - {1 \over 3}} \right] + \left[ { - {1 \over 3} - {1 \over {100}}} \right] + \left[ { - {1 \over 3} - {2 \over {100}}} \right] + .... + \left[ { - {1 \over 3} - {{99} \over {100}}} \right]$ is :
A.
- 153
B.
- 135
C.
- 133
D.
- 131
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6= – 48, then S10 is equal to :
A.
- 320
B.
- 380
C.
- 460
D.
- 210
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
Let a1, a2, a3,......be an A.P. with a6 = 2. Then the common difference of this A.P., which maximises the product a1a4a5, is :
A.
${3 \over 2}$
B.
${6 \over 5}$
C.
${8 \over 5}$
D.
${2 \over 3}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
The sum
$1 + {{{1^3} + {2^3}} \over {1 + 2}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 2 + 3}} + ...... + {{{1^3} + {2^3} + {3^3} + ... + {{15}^3}} \over {1 + 2 + 3 + ... + 15}}$$ - {1 \over 2}\left( {1 + 2 + 3 + ... + 15} \right)$ is equal to :
A.
620
B.
1240
C.
1860
D.
660
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
Let $a$, b and c be in G.P. with common ratio r, where $a$ $ \ne $ 0 and 0 < r $ \le $ ${1 \over 2}$ . If 3$a$, 7b and 15c are the first three terms of an A.P., then the 4th term of this A.P. is :
A.
$a$
B.
${7 \over 3}a$
C.
5$a$
D.
${2 \over 3}a$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
If a1, a2, a3, ............... an are in A.P. and a1 + a4 + a7 + ........... + a16 = 114, then a1 + a6 + a11 + a16 is equal to :
A.
38
B.
98
C.
76
D.
64
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
The sum
${{3 \times {1^3}} \over {{1^3}}} + {{5 \times ({1^3} + {2^3})} \over {{1^2} + {2^2}}} + {{7 \times \left( {{1^3} + {2^3} + {3^3}} \right)} \over {{1^2} + {2^2} + {3^2}}} + .....$ upto 10 terms is:
A.
600
B.
660
C.
680
D.
620
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are addded to the total number of balls used in forming the equilaterial triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is :-
A.
262
B.
190
C.
157
D.
225
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11th term is :-
A.
–25
B.
–36
C.
25
D.
–35
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 +.... upto 11th term is :-
A.
945
B.
916
C.
915
D.
946