JEE Main Mathematics - Sequences and Series
Exam Duration: 60 Mins Total Questions : 30
The value of \(0.4\bar { 23 } \) is
- (a)
\(419\over 999\)
- (b)
\(419\over 990\)
- (c)
\(423\over 1000\)
- (d)
NONE OF THESE
If log 2, log (2x-1) and log (2x+3) are in A.P., then 2, 2x-1, 2x+3 are in
- (a)
A.P.
- (b)
Their reciprocals in A.P.
- (c)
G.P.
- (d)
NONE OF THESE
If x, y, z are positive integers then (x+y)(y+z)(z+x) is
- (a)
<8xyz
- (b)
=8xyz
- (c)
>8xyz
- (d)
NONE OF THESE
If x1, x2, x3as well as y1, y2, y3 are in G.P. with the same common ratio then the points (x1, y1), (x2 y2) and (x3, y3)
- (a)
lie on a straight line
- (b)
lie on the ellipse
- (c)
lie on a circle
- (d)
are verticles of a triangle
If a1, a2, a3,..., an are non-zero real numbers such that \(({ a }_{ 1 }^{ 2 }\quad +\quad { a }_{ 2 }^{ 2 }\quad +\quad { a }_{ 3 }^{ 2 }\quad +...+\quad { a }_{ n-1 }^{ 2 })\quad ({ a }_{ 2 }^{ 2 }\quad +\quad { a }_{ 3 }^{ 2 }\quad +...+{ a }_{ n }^{ 2 })\quad \le \quad (a_1a_2 \ +\ a_2a_3\ + ...a_{n-1}a_n)^2\)
then \(a_1,a_2,a_3,...a_n\) are in
- (a)
AP
- (b)
G.P.
- (c)
H.P.
- (d)
NONE OF THESE
If a, b, c are in H.P. then
- (a)
an + cn > bn
- (b)
\(a^n\ +\ c^n\ > 2b^n\)
- (c)
\(a^n\ +\ c^n\ > (2b)^n\)
- (d)
NONE OF THESE
If \(a_1,a_2,a_3, . . . a_n\) are in A.P. where an>0 for all n then value of \(\frac { 1 }{ \sqrt { { a }_{ 1 } } +\sqrt { { a }_{ 2 } } } +\frac { 1 }{ \sqrt { { a }_{ 2 } } +\sqrt { { a }_{ 3 } } } +...+\frac { 1 }{ \sqrt { { a }_{ n-1 } } +\sqrt { { a }_{ n } } } \) is
- (a)
\(\frac { 1 }{ \sqrt { { a }_{ 1 } } +\sqrt { { a }_{ n } } } \)
- (b)
\(\frac { 1 }{ \sqrt { { a }_{ 1 } } -\sqrt { { a }_{ n } } } \)
- (c)
\(\frac { n }{ \sqrt { { a }_{ 1 } } +\sqrt { { a }_{ n } } } \)
- (d)
\(\frac { n-1 }{ \sqrt { { a }_{ 1 } } +\sqrt { { a }_{ n } } } \)
If \(\frac { { e }^{ x } }{ 1-x } \quad =\quad { B }_{ 0 }+{ B }_{ 1 }x+{ B }_{ 2 }{ x }^{ 2 }+...+{ B }_{ n }{ x }^{ n }+...,\) then value of \(B_n -B_{n-1}\) is
- (a)
1
- (b)
\(1\over n\)
- (c)
\(1\over n!\)
- (d)
NONE OF THESE
The sum of the series, \(\frac { 1 }{ { n }^{ 2 } } +\frac { 1 }{ { 2n }^{ 4 } } +\frac { 1 }{ { 3n }^{ 6 } } +...\) is
- (a)
\(log\left( \frac { { n }^{ 2 } }{ { n }^{ 2 }+1 } \right) \)
- (b)
\(log\left( \frac { { n }^{ 2 }+1 }{ { n }^{ 2 } } \right) \)
- (c)
\(log\left( \frac { { n }^{ 2 } }{ { n }^{ 2 }-1 } \right) \)
- (d)
NONE OF THESE
If x, Y and z are positive integers, then value of expression ( X + Y) (Y + Z) (Z + X) is
- (a)
= 8 xyz
- (b)
> 8 xyz
- (c)
< 8 xyz
- (d)
= 4 xyz
If X, Y and Z are in AP and tan-1X, tan-1Y and tan-1Z are also in AP, then
- (a)
x = y = z
- (b)
2x = 3y = 6z
- (c)
6x = 3y = 2z
- (d)
6x = 4y = 3z
If ab2c3, a2b3c4 a3b4c5 are in AP (a, b, c > 0), then the minimum value of a + b + c is
- (a)
1
- (b)
3
- (c)
5
- (d)
9
If \(\sum _{ i=1 }^{ 21 }{ { a }_{ i }=693 } \) , where a1,a2,.....,a21 are in AP, then the value of \(\sum _{ r=0 }^{ 10 }{ { a }_{2r+1 } } \) is
- (a)
361
- (b)
363
- (c)
365
- (d)
398
If a, b, c are in AP, then \(\frac { 1 }{ \sqrt { b } +\sqrt { c } } ,\frac { 1 }{ \sqrt { c } +\sqrt { a } } ,\frac { 1 }{ \sqrt { a } +\sqrt { b } } \) are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
no definite sequence
If (1.05)50= 11.658, thin \(\sum _{ n=1 }^{ 49 }{ ({ 1.05) }^{ n } } \) equals
- (a)
208.34
- (b)
212.12
- (c)
212.16
- (d)
213.16
If a, b, c are digits, then the rational number represented by 0. cababab ... is
- (a)
cab/990
- (b)
(99c + ba)/990
- (c)
(99c + 10a + b)/ 99
- (d)
(99c + 10a +b )/ 990
If < an > and < bn > be two sequences given by an = (x)1/2n + (y)1/2n and bn = (x)1/2n -(y )1/2n for all n \(\in \) N, then a1 a2 a3 ... an is
- (a)
\(\frac { x+y }{ { b }_{ n } } \)
- (b)
\(\frac { x-y }{ { b }_{ n } } \)
- (c)
\(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ { b }_{ n } } \)
- (d)
\(\frac { { x }^{ 2 }-{ y }^{ 2 } }{ { b }_{ n } } \)
If a sequence or series is not a direct form of an AP, GP, etc Then its nth term can not be determined. In such cases, we use the following steps to find the nth term (Tn) of the given sequence.
Step - I: Find the differences between the successive terms of the given sequence. If these differences are in AP, then take Tn = an2+ bn + c, where a, b, c are constants.
Step-II: If the successive differences finding in step I are in GP with common ratio r, then take Tn = a + bn + cr n-1, where a, b, c are constants.
Step - Ill : If the second successive differences (Differences of the differences) in step I are in AP, then take Tn = an3 + bn2 + cn + d, where a, b, c, d are constants.
Step-Iv : If the second successive differences (Differences of the differences) in step I are III GP, then take Tn = an2 + bn + c + dr n-1 ,where a, b, c, d are constants.
Now let sequences:
A: 1,b, 18,40,75,126, ...... B : 1,1,6,26,91,291,..... C : In 2 In 4, In 32, In 1024,......
The format of nth term (Tn) of the sequence C is
- (a)
an2 + bn + c
- (b)
an3 + bn2 + cn + d
- (c)
an + b + crn-1
- (d)
an2 + b + c + drn-1
We are giving the concept of arithmetic mean of mth power, Let a, b > 0 and a \(\neq \) b and let m be a real number. Then
\(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } >{ \left( \frac { a+b }{ 2 } \right) }^{ m },\quad if\quad m\in R\sim [0,1]\)
However if \(m\in (0,1),\) then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } <{ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
Obviously if m \(\in \{ 0,\quad 1\} ,\)then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } ={ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
If a, b be positive and a + b = 1 (a \(\neq \) b) and if A= \(\sqrt [ 3 ]{ a } +\sqrt [ 3 ]{ b } \) then the correct statement is
- (a)
A > 22/3
- (b)
\(A=\frac { { 2 }^{ 2/3 } }{ 3 } \)
- (c)
A < 22/3
- (d)
A = 22/3
The income of a person is Rs. 3,00,000, in the first year and he receives an increase of Rs. 10,000 to his income per tear for the next 19 years. Find the total amount, he received in 20 years.
- (a)
Rs. 7800000
- (b)
Rs. 7900000
- (c)
Rs. 7700000
- (d)
Rs. 790000
If x, y, z are in A.P., then\({1\over \sqrt{y}+\sqrt{z}},{1\over \sqrt{z}+\sqrt{x}},{1\over \sqrt{x}+\sqrt{y}}\) are in
- (a)
A.P.
- (b)
G.P.
- (c)
A.G.P.
- (d)
no definite sequence
In a G.P., the 3rd term is 24 and the 6th term is 192. Find the 10th term.
- (a)
3076
- (b)
3071
- (c)
3072
- (d)
3074
if a,b,c,d are in G.P., then \({(a^2+b^2+c^2)(b^2+c^2+d^2)\over (ab+bc+cd)^2}=\)
- (a)
1
- (b)
2
- (c)
3
- (d)
5
Find the sum of first n terms and the sum of first 5 terms of the geometric series 1+\({2\over3}+{4\over 9}+..\)
- (a)
\(3[1-({2\over3})^n],{211\over81}\)
- (b)
\([1-({2\over3})^n],{210\over81}\)
- (c)
\((1+({2\over3})^n),{212\over81}\)
- (d)
None of these
How many terms of the geometric series 1+4+16+64+.. will make the sum 5461?
- (a)
6
- (b)
7
- (c)
8
- (d)
9
Find the G.M. between 4 and 9.
- (a)
6
- (b)
8
- (c)
9
- (d)
12
The minimum value of 8sin(x/8)+8cos(x\8) is
- (a)
\(2^{1\over {3-\sqrt{2}\over \sqrt{2}}}\)
- (b)
\(2^{3+\sqrt{2}\over \sqrt{2}}\)
- (c)
\(2^{1\over {3+\sqrt{2}\over \sqrt{2}}}\)
- (d)
\(2^{3-\sqrt{2}\over \sqrt{2}}\)
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is
- (a)
3
- (b)
\(1\over3\)
- (c)
2
- (d)
\(1\over2\)
Let Sn denote the sum of the cubes of the first n natural numbers and sn denote the sum of the first n natural numbers. Then \(\sum^{n}_{r=1}{S_r\over S_r}\) equals
- (a)
\({n(n+1)(n+2)\over6}\)
- (b)
\({n(n+1)\over2}\)
- (c)
\({n^3+3n+2\over2}\)
- (d)
None of these
Statement-I : There are infinite geometric progressions for which 27, 8 and 12 are three of its terms (not necessarily consecutive).
Statement-II : Given terms are integers.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -1.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.