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

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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