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m17 q25

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m17 q25 [#permalink] New post 06 Jul 2010, 06:30
Series A(n) is such that i*A(i) = j*A(j) for any pair of positive integers (i,j) . If A(1) is a positive integer, which of the following is possible?

I. 2A(100) = A(99) + A(98)
II. A(1) is the only integer in the series
III. The series does not contain negative numbers

a. I only
b. II only
c. I and III only
d. II and III only
e. I, II, and III

[Reveal] Spoiler:
II is possible. If A(1)=1 then A(k) = \frac{1}{k} . There are no more integers in the sequence except A(1) .
III is possible as well. Same example applies.
I is not possible. Because i*A(i) = (i + 1)A(i + 1)A(i + 1) = \frac{i}{i + 1} A(i) which is less then A(i). This means that this sequence is a decreasing sequence in which every subsequent element is smaller than its predecessor. Thus, A(100) + A(100) \lt A(99) + A(98)
The correct answer is D.


Can someone please explain this?
I have no clue what data is provided and what answer is expected in this question
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Re: m17 q25 [#permalink] New post 08 Jul 2010, 17:01
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Expert's post
yeahwill wrote:
Series A(n) is such that i*A(i) = j*A(j) for any pair of positive integers (i,j) . If A(1) is a positive integer, which of the following is possible?

I. 2A(100) = A(99) + A(98)
II. A(1) is the only integer in the series
III. The series does not contain negative numbers

a. I only
b. II only
c. I and III only
d. II and III only
e. I, II, and III

[Reveal] Spoiler:
II is possible. If A(1)=1 then A(k) = \frac{1}{k} . There are no more integers in the sequence except A(1) .
III is possible as well. Same example applies.
I is not possible. Because i*A(i) = (i + 1)A(i + 1)A(i + 1) = \frac{i}{i + 1} A(i) which is less then A(i). This means that this sequence is a decreasing sequence in which every subsequent element is smaller than its predecessor. Thus, A(100) + A(100) \lt A(99) + A(98)
The correct answer is D.


Can someone please explain this?
I have no clue what data is provided and what answer is expected in this question


This question was posted in PS forum. Below is my post from there:

A set of numbers a_1, a_2, a_3, ... have the following properties: i*a_i=j*a_j and a_1=positive \ integer, so 1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer.

We should determine whether the options given below can occur (note that the question is which can be true, not must be true).

I. 2a_{100}=a_{99}+a_{98} --> as 100a_{100}=99a_{99}=98a_{98}, then 2a_{100}=\frac{100}{99}a_{100}+\frac{100}{98}a_{100} --> reduce by a_{100} --> 2=\frac{100}{99}+\frac{100}{98} which is not true. Hence this option cannot be true.

II. a_1 is the only integer in the series. If a_1=1, then all other terms will be non-integers --> a_1=1=2a_2=3a_3=... --> a_2=\frac{1}{2}, a_3=\frac{1}{3}, a_4=\frac{1}{4}, and so on. Hence this option can be true.

III. The series does not contain negative numbers --> as given that a_1=positive \ integer=n*a_n, then a_n=\frac{positive \ integer}{n}=positive \ number, hence this option is always true.

Answer: D (II and III only).

Hope it's clear.
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Re: m17 q25 [#permalink] New post 13 Jul 2012, 11:22
Bunuel,
I agree with (i) and (iii). However, I am not sure about (ii).

Why did you substitute a1 =1 ? If A(1) is the only integer => n=1; But how do we know that a1 = 1? a1 could be anything....a1=2 also holds good because there is only one number. Correct? Essentially, there is no A(2), A(3) etc.

Thoughts?
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Re: m17 q25 [#permalink] New post 13 Jul 2012, 11:28
Expert's post
voodoochild wrote:
Bunuel,
I agree with (i) and (iii). However, I am not sure about (ii).

Why did you substitute a1 =1 ? If A(1) is the only integer => n=1; But how do we know that a1 = 1? a1 could be anything....a1=2 also holds good because there is only one number. Correct? Essentially, there is no A(2), A(3) etc.

Thoughts?


The question asks "which of the following is possible" or which of the following COULD be true. So, we don't know that a_1=1, but a_1 COULD be 1 and in this case it would be the only integer in the sequence. So, II is certainly POSSIBLE.

Hope it's clear.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: m17 q25   [#permalink] 13 Jul 2012, 11:28
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