07 - SEQUENCES AND SERIES
Page 1
( Answers at he end of all questions )
∑
If x =
∞
an ,
∑
y =
n=0
∑ cn ,
n=0
where a, b, c are in A.P. and
n=0
l a l < 1,
l b l < 1,
l c l < 1, then x, y, z are in
( a ) G.P.
( b ) A.P.
( c ) Arithmetic-Geometric Progression
e - 1 e
∑
n
1 n 2
and
n
r =0
1 n - 1 2
d nf. is
e + 1
[ AIEEE 2005 ]
then
tn Sn
(d)
2n - 1 2
,
C
1
=
[ AIEEE 2004 ]
m
(c) n
[ AIEEE 2005 ]
2 e
r
∑
tn =
Cr
(b)
(d)
2 e n
1
r =0
(a)
e - 1
(c)
e n
If S n =
1 1 1 + + + ……… 4⋅ 2! 16 ⋅ 4 ! 64 ⋅ 6 !
e + 1
(b)
( d ) H.P.
ce .c
The sum of the series 1 +
(a)
(3)
∞
z =
ra
(2)
bn ,
om
∞
(1)
Let T r be the rth term of an A.P
whose first term is a and common difference is 1 1 d. If for some positive integers m, n, m ≠ n, T m = and T n = , then n m 1 1 1 (a) 0 (b) 1 (c) (d) + [ AIEEE 2004 ] mn m n
(5)
The sum of the f rst n terms of he series
w w
.e
xa
(4)
1
2
+ 2
2
+ 3
2
+
2
⋅ 42
+
5
2
+
2
⋅ 62
+ ….. is
n ( n + 1 )2 2
when n is
w
ven. When n is odd, the sum is
(6)
(a)
3 n ( n + 1) 2
(b)
n2 ( n + 1 ) 2
(c)
n ( n + 1 )2 4
(d)
n ( n + 1) 2
The sum of the series
(a)
e2 - 1 2
(b)
2
[ AIEEE 2004 ]
1 1 1 + + + ..... is 4! 6! 2!
( e - 1 )2 2e
(c)
e2 - 1 2e
(d)
e2 - 2 e
[ AIEEE 2004 ]
07 - SEQUENCES AND SERIES
Page 2
( Answers at he end of all questions )
2⋅ 3
+
1
- ...... ∞
3⋅ 4
( c ) log e 2 - 1
( b ) 2log e 2
is
( d ) log e
2
4 e
[ AIEEE 2003 ]
( b ) G. P.
ce .c
If the sum of the roots of the quadratic equation ax + bx + c = 0 is equal to the sum a b c of the squares of their reciprocals, then , , are in c a b ( a ) A. P.
(9)
1
om
( a ) log e 2
(8)
1 1⋅ 2
The sum of the series
( c ) H. P.
( d ) A. G. P
[ AIEEE 2003 ]
The value of 1. 2. 3 + 2. 3. 4 + 3. 4. 5 + … + n terms is (a)
(b)
(d)
m
(c)
n ( n + 1) ( n + 2 ) ( n + 3 ) 12 n ( n + 1) ( n + 2 ) ( n + 3 ) 4
n(n + 1 (n + 2)(n + 3) 3 n + 2)(n + 3)(n + 4 ) 6
ra
(7)
[ AIEEE 2002 ]
xa
( 10 ) If the third term of an A. P. s 7 and its 7th term is 2 more than three times of its third term, then the sum o its f st 20 terms is ( b ) 74
( c ) 740
.e
( a ) 228
( 4 ) 1090
[ AIEEE 2002 ]
( 11 ) An infinite G. P. h s first term ‘ x ’ and sum 5, then
w w
( a ) x ≥ 10
( c ) x < - 10
( b ) 0 < x < 10
( 12 ) If a1, a2, ….. an
w ( 13 )
1/n
(b) (n + 1)c
1/n
- 1
+ 2a n is
( c ) 2n c
1/n
2
Suppose
2
( d ) ( n + 1 ) ( 2c )
2
a, b, c are in A. P. and a , b , c , are in G. P. 3 a + b + c = , then the value of a is 2 (a)
1 2 2
{ IIT 2004 }
are positive real numbers whose product is a fixed number c, then
the minimum value of a1 + a2 + … + a n ( a ) n ( 2c )
( d ) - 10 < x < 0
(b)
1 2 3
(c)
1 2
1 3
(d)
1 2
1 2
If
1/n
[ IIT 2002 ]
a < b < c
and
[ IIT 2002 ]
07 - SEQUENCES AND SERIES
Page 3
( Answers at he end of all questions )
If the sum of the first 2n terms of the A. P. 2, 5, 8, ….. , is equal to the sum of the first n terms of the A. P. 57, 59, 61, ….. , then n equals ( a ) 10
( b ) 12
( c ) 11
( d ) 13
[ IIT 2001 ]
om
( 14 )
( 15 ) If the positive numbers a, b, c, d are in A. P., then abc, abd, acd, bcd ar ( b ) in A. P.
( c ) in G. P.
(d) i
( 16 ) If a, b, c, d are positive real numbers such th t M = ( a + b ) ( c + d ) satisfies the relation
3 7
( b ) 2,
.e (b) 3
(c) 5
3 , 2
(c)
Let a1, a2, …, a10 be n A. P. and a10 = h10 = 3, then 4 h 7 is
w w w
3 8
xa
4 , 7
(a) 2
( 19 )
a + b + c + d = 2, then
(d) 3 ≤ M ≤ 4
[ IIT 2000 ]
Consider an infinite geometric series wi h first erm a and common ratio r. If its sum 3 is 4 and the second term is , then a nd r are 4 (a)
( 18 )
(c) 2 ≤ M ≤ 3
m
( 17 )
(b) 1 ≤ M ≤ 2
[ IIT 2001 ]
ra
(a) 0 ≤ M ≤ 1
H P.
ce .c
( a ) not in A. P. / G. P. / H. P.
( d ) 3,
1 4
[ IIT 2000 ]
h1, h2, …, h10 be in H. P. If a1 = h1 = 2 and
(d) 6
for a positive integer n, a ( n ) = 1 +
( a ) a ( 100 ) ≤ 100 ( c ) a ( 200 ) ≤ 100
1 2
[ IIT 1999 ]
1 1 1 1 , then + + + ..... + n 2 3 4 2 - 1
( b ) a ( 100 ) > 100 ( d ) a ( 200 ) > 100
[ IIT 1999 ]
( 20 ) Let Tr be the rth term of an A. P., for r = 1, 2, 3, … If for some positive integers 1 1 m, n, we have Tm = and Tn = , then Tm n equals n m (a)
1 mn
(b)
1 1 + m n
(c) 1
(d) 0
[ IIT 1998 ]
07 - SEQUENCES AND SERIES
Page 4
( Answers at he end of all questions )
( 21 ) If x > 1, y > 1, z > 1 are in G. P., then ( b ) H. P.
( c ) G. P.
1 , 1 + ln y
1 1 + ln z
are in
( d ) None of these
[ IT 1998 ]
om
( a ) A. P.
1 , 1 + ln x
( a ) 127
( b ) 63
m
+ 1 ) divides
ce .c
( 22 ) If n > 1 is a positive integer, then the largest integer m such that ( n 2 127 ( 1 + n + n + ….. + n ) is ( c ) 64
( d ) 32
[ IIT 1995 ]
( 23 ) The product of n positive numbers is unity. Then thei sum is ( b ) divisible by n
( 24 ) The sum of n terms of the se es n
(b) 1 - 2-
xa
(a) 2
- n - 1
[ IIT 1991 ]
ra
( d ) never less than n
m
( a ) a positive integer 1 ( c ) equal to n + n
n
1 3 7 15 + + + + ..... 2 4 8 16
(c) n + 2-
n
+ 1
(d) 2
n
is equal to
- 1
[ IIT 1988 ]
.e
( 25 ) If the first and he ( 2n - 1 )th terms of an A. P., G. P. and H. P. are equal and their nth terms are a, , c respectively, then
w w
(a) a = b = c
(b) a ≥ b ≥ c
(c) a + c = b
2
( d ) ac - b = 0
[ IIT 1988 ]
w
( 26 ) If a, , c, d and p are distinct real numbers such that 2 2 2 2 2 2 ( + b + c ) p - 2 ( ab + bc + cd ) p + ( b + c + d ) ≤ 0, then a, b, c and d ( a ) are in A. P.
( b ) are in G. P.
( c ) are in H. P.
( 27 ) If a, b, c are in G. P., then the equations ax d e f , , have a common root if are in a b c ( a ) AP
( b ) GP
( c ) HP
2
( d ) satisfy ab = cd
+ 2bx + c = 0 and dx
( d ) none of these
2
[ IIT 1987 ]
+ 2ex + f = 0
[ IIT 1985 ]
07 - SEQUENCES AND SERIES
Page 5
( Answers at he end of all questions )
( 28 ) The third term of a geometric progression is 4. The product of the first five terms is 3
(b) 4
5
(c) 4
4
( d ) none of these
[ IIT 1982 ]
om
(a) 4
( 29 ) If x1, x2, ….. , xn are any real numbers and n is any positive integer, t en
∑
(a) n
xi2
i=1 n
∑
(c)
2 n < xi i=1
n
∑
(b)
i=1
2 n ≥ n xi i=1
∑
x i2
i=1
∑
1+
1 1- x
, x
are consecutive terms of a series in
x
( c ) A. P.
( d ) A. P., G. P.
1 n ( n - 1 ) Q, where Sn denotes the sum of the first n terms of an 2 then the common difference is
w w w
1-
( b ) G. P.
a) P + Q
3
( b ) 2P + 3Q
( c ) 2Q
(d) Q
2
If Sn = n + n + n + 1, where Sn denotes the sum of the first n terms of a series and t m = 291, then m = ( a ) 10
( 34 )
[ IIT 1979 ]
If Sn = n P + A. P
( 33 )
( d ) none of these
.e
( a ) H. P.
( 32 )
(c) 1
and
respectively of an A. P. and also of a
m
1
erm
[ IIT 1982 ]
is equal to
xa
( 31 )
(b) 0
∑
ra
xy - z yz - x zx - y
( a ) xyz
2 n ≥ xi i=1
( d ) none of these
( 30 ) If x, y and z are the p th, q th and r th G. P., then
x i2
ce .c
n
( b ) 11
( c ) 12
( d ) 13
If the first term minus third term of a G. P. = 768 and the third term minus seventh term of the same G. P. = 240, then the product of first 21 terms = (a) 1
(b) 2
(c) 3
(d) 4
07 - SEQUENCES AND SERIES
Page 6
( Answers at he end of all questions )
( 35 ) If the sequence a1, a2, a3, … an form an A. P., a1
- a22 + a32 - … + a2n – 12 - a2n2 =
n ( a 12 - a 2n 2 ) 2n - 1
(b)
2n ( a 2n 2 - a 12 ) n - 1
(c)
n ( a 12 + a 2n 2 ) n + 1
( d ) None of these
( 36 ) If Tr denotes rth term of an H. P. and (b) 6
(c) 7
(d) 8
T1 - T4 = 7, t en T6 - T9
T2 - T5 T11 - T8
=
ra
(a) 5
ce .c
(a)
om
2
then
( 37 ) The sum of any ten positive real numbers multiplied by the sum of their reciprocals is ( c ) ≥ 100
( d ) ≥ 200
If S n denotes the sum of ir t n terms of an A. P. and S 2n = 3 S n , then the ratio S 3n is equal to Sn
xa
( 38 )
( b ) ≥ 50
m
( a ) ≥ 10
(b) 6
.e
(a) 4
(c) 8
( d ) 10
w w
( 39 ) If a, b, c a e three unequal positive quantities in H. P., then 10
10
10
20
20
( b ) a + c < 2b ( d ) none of these
w
(a) a c < 2b 3 3 3 (c) + c < 2b
20
Answers
1 d
2 d
3 a
4 a
5 b
6 b
7 d
8 c
9 c
10 c
11 b
12 a
13 d
14 c
15 d
16 a
17 d
18 d
19 a,d
20 c
21 b
22 c
23 d
24 c
25 b,d
26 b
27 a
28 b
29 d
30 c
31 c
32 d
33 a
34 a
35 a
36 b
37 c
38 b
39 d
40
