1.Algebra Booster

Sequence and Series 1.9
53. Which term of the GP 2, 1, 1 , 1 ,
2 4 º is 1
128 ?
54. If the 2nd and 5th terms of a GP are 24 and 81, respectively,
find the GP.
55. If the 3rd term of a GP is 3, then find the product of its
first five terms.
56. The sum of three numbers in GP is 21 and the sum of
their squares is 189, find the numbers.
57. If the first term of a GP is 1 and the sum of the 5th and
1st term is 82, find the common ratio.
58. If the pth, qth and rth terms of a GP are a, b and c, respectively,
prove that
a q–r ◊ b r–p ◊ c p–q = 1
59. If the first and the nth terms of a GP are a and b, respectively
and if P is the product of the first n terms, prove
that P 2 = (ab) n .
60. The (m + n)th and (m – n)th terms of a GP are p and q,
respectively. Show that the mth and nth terms are pq
and
Ê q ˆ
pÁ
Ë p
˜
¯
m/2n
, respectively.
61. If a, b, c, d and p are different real numbers such that
(a 2 + b 2 + c 2 )p 2 – 2(ab + bc + cd)p + (b 2 + c 2 + d 2 ) = 0,
show that a, b, c and d are in GP.
62. If a, b, c are respectively the pth, qth and rth terms of a
GP, show that
(q – r) log a + (r – p) log b + (p – q) log c = 0
63. If (1 – k)(1 + 2x + 4x 2 + 8x 3 + 16x 4 + 32x 5 ) = 1 – k 6 ,
where k π 1, find the value of k x .
64. If a and b be the roots of x 2 – 3x +a = 0 and g and d be
the roots of x 2 – 12x + b = 0 and numbers a, b, g, d (in
order) form an increasing GP, prove that the value of
a = 2 and b = 32.
65. If three distinct real numbers x, y, z are in GP such that
x + y + z = ax, find the value of a.
SUM OF N TERMS OF GP
66. Find the sum of 1 + 3 + 9 + 27 + … to n terms.
1 1 1
67. Find the sum of 1+ + + +º ton
terms.
2 4 8
68. How many terms of the series 1 + 3 + 3 2 + 3 3 + … must
be taken to make 3280?
69. Find the sum of the geometric series
(x + y) + (x 2 + xy + y 2 ) + (x 3 + x 2 y + xy 2 + y 3 ) + … to n-
terms.
70. Find the sum of the geometric series
a + a + a + … +
a .
2 3
(1 + i) (1 + i) (1 + i) (1 + i) n
k
71. Find the value of  (2 + 3 ) .
10
k = 1
n-1
n
72. Evaluate  (2 + 3 ) .
73. Evaluate
(i)
(ii)
n
Â
k = 1
n
Â
n=
1
n
n=
1
k- 1 k+
1
(4 + 5 )
ÊÊ1ˆ
Ê1ˆ
ÁÁ ˜ + Á ˜
ËË3¯
Ë5¯
n- 1 n+
1
74. Find the sum of the following series
(i) 5 + 55 + 555 + … to n terms
(ii) 7 + 77 + 777 + … to n terms
(iii) 9 + 99 + 999 + … to n terms
75. Find the sum of (6666 … 6) 2 + ( 8888 … 8) 2 (upto n
digits).
10 1 1
76. Find the sum  È
n- n+
Ê1ˆ Ê1ˆ
˘
ÍÁ ˜ + Á ˜ ˙
n=
1ÎË2¯ Ë5¯
˚
.
77. Prove that the sum to n terms of the series 11 + 103 +
ˆ
˜
¯
1005 +… is 10 9 (10n – 1) + n 2 .
78. Find the least value of n for which the sum 1 + 3 + 3 2 +
… to n terms is greater than 7000.
79. Find the sum of n terms of the series
1 5 19 65
+ + + + …to n terms .
3 9 27 81
2 8 26 30
80. If S = + + + + … to n terms, find the value
3 9 27 81
of S.
81. If S be the sum, P the product and R the sum of the
n
S 2
reciprocals of n terms of a GP, prove that
Ê Á
ˆ ˜ = P .
ËR¯
82. If f is a function satisfying f(x + y) = f(x) + f(y) for all
x, y ΠN such that f(1) = 3 and
n
Â
x=
1
f( x) = 120 , find the
value of n.
83. Let a n
be the nth term of the GP of positive numbers.
Let
100
 a2n
= a and
n=
1
100
 a2n-1=
b such that a π b.
n=
1
Prove that the common ratio of the GP is a b .
SUM OF AN INFINITE GP
84. Find the sum of
( 2 1) 1 ( 2 1) …to
85. Find the sum of
1 1 1 1 1 1
2
2
3
3
2
4
3
5
2
6
3