1.Algebra Booster

Sequence and Series 1.21
Passage VI
Suppose s 1
, s 2
,…, s n
are the sum of n geometric series of infinite
terms, whose first terms are 1, 2, 3, …, n and the common
ratios 1 , 1 ,…
1
2 3 n + 1
, respectively.
(i) If s 1
+ s 2
+ … + s n
= 7, the value of n is
(a) 12 (b)* 13 (c) 11 (d) 14
2 2 2
(ii) The value of s1 + s2 +º+ s2n
- 1 is
(a) n(2n + 1)(4n + 1) – 1
1
(b) (2 1)(4 1) 1
3 n n+ n+
-
(c)
(d)
1 (2 1)(4 1)
3 n n+ n+
1 (2 1)(4 1) 2
3 n n+ n+
-
3 3 3
(iii) If s1 + s2+º+ s2k
- 1= 1800 , the value of k is
(a) 15 (b) 16 (c) 5 (d) 6
Passage VII
Let A 1
, A 2
, A 3
, …, A n
be the arithmetic means between –2 and
1027 and G 1
, G 2
, G 3
, …, G n
be geometric means between 1
and 1024. The product of geometric means is 2 45 and the sum
of the arithmetic means is 1025 ¥ 171.
On the basis of above information, answer the following
questions:
(i) The value of n is
(a) 7 (b) 9 (c) 11 (d) None
(ii) The value of m is
(a) 340 (b) 342 (c) 344 (d) 346
(iii) The value of G 1
+ G 2
+ G 3
+ … + G n
is
(a) 1022 (b) 2044 (c) 512 (d) None
(iv) The common difference of the progression A 1
, A 3
, A 5
,
…, A m–1
is
(a) 6 (b) 3 (c) 2 (d) 1
(v) The numbers 2A 171
, (G 5
) 2 + 1 and 2A 172
are in
(a) AP (b) GP (c) HP (d) AGP
Passage VIII
There are two sets A and B each of which consists of three
numbers in AP, whose sum is 15 where D and d are the common
differences such that D – d = 1. If
p 7
q = , where p and
8
q are the products of the numbers, respectively, and d > 0, in
two sets.
On the basis of the above information, answer the following
questions:
(i) The value of p is
(a) 100 (b) 120 (c) 105 (d) 110.
(ii) The value of q is
(a) 100 (b) 120 (c) 105 (d) 110
(iii) The value of D + d is
(a) 1 (b) 2 (c) 3 (d) 4
Passage IX
Four different integers form an increasing AP. One of these
numbers is equal to the sum of the squares of the other three
numbers.
On the basis of the above information, answer the following
questions:
(i) The smallest number is
(a) –2 (b) 0 (c) –1 (d) 2
(ii) The common difference of the four numbers is
(a) 2 (b) 1 (c) 3 (d) 4
(iii) The sum of the four numbers is
(a) 10 (b) 8 (c) 2 (d) 6
Matrix Match
(For JEE-Advanced Examination Only)
1. Match the following columns:
(A)
(B)
(C)
(D)
Column I
If a 2 , b 2 , c 2 are in AP,
b+ c c+ a a + b
, , are in
a b c
If a, b, c are in HP,
a b c
, , are in
b+ c c+ a a + b
If a, b, c are in AP as well
as in GP, a 2 , b 2 , c 2 are in
If b + c, c + a, a + b are in
a b c
HP, , ,
b+ c c+ a a + b
are in
2. Match the following columns:
(P)
(Q)
(R)
(S)
Column II
AP
GP
HP
AGP
Column I
Column II
(A) If 2, A 1
, A 2
, A 3
, A 4
, 8 are in AP, the (P) 2
value of A 1
+ A 2
+ A 3
+ A 4
is
(B) If 2, G 1
, G 2
, G 3
, G 4
, G 5
, 16 are in (Q) 5/2
GP, the value of
G 1
◊ G 2
◊ G 3
◊ G 4
◊ G 5
is
(C) If 2, H 1
, H 2
, H 3
, H 4
, H 5
, H 6
,
3 are in HP, the value of
1 1 1 1 1 1
+ + + + +
H1 H2 H3 H4 H5 H6
is
(R) 20
3. Match the following columns:
(A)
Column I
If x 1
, x 2
, x 3
, …, x 10
are in HP,
xx 1 2+ x2x3+ xx 3 4+º+
xx 1 10
xx
is
1 10
Column II
(P) 60