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Re: Series A(n) is such that i*A(i) = j*A(j) [#permalink]
07 Feb 2012, 03:02
1
This post received KUDOS
Expert's post
vinnik 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. 2*A(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 & III only D) II & III only E) I, II & III
Will appreciate if anyone explains this question with an easy method.
Thanks & Regards Vinni
Probably it should be sequence instead of series.
A sequence 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 (notice that the question is which of the following COULD be true, not MUS 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.
Re: Series A(n) is such that i*A(i) = j*A(j) [#permalink]
07 Feb 2012, 03:34
Bunuel wrote:
A sequence of numbers \(a_1\), \(a_2\), \(a_3\), ... have the following properties: \(i*a_i=j*a_j\) and \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\).
Bunuel,
I didnt get this part. I seem to misunderstood the q.stem
could u please clarify it? _________________
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Re: Series A(n) is such that i*A(i) = j*A(j) [#permalink]
07 Feb 2012, 03:39
3
This post received KUDOS
Expert's post
LalaB wrote:
Bunuel wrote:
A sequence of numbers \(a_1\), \(a_2\), \(a_3\), ... have the following properties: \(i*a_i=j*a_j\) and \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\).
Bunuel,
I didnt get this part. I seem to misunderstood the q.stem
could u please clarify it?
Sure.
Given: \(a_1=positive \ integer\). Next, \(i*a_i=j*a_j\), notice that we have the same multiple and the same index of a on both sides: \(1*a_1=2*a_2\), \(2*a_2=3*a_3\), \(a_3=4*a_4\).... Hence, \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\) (it equal to an integer since \(a_1=positive \ integer\)).
Re: Series A(n) is such that i*A(i) = j*A(j) for any pair of [#permalink]
17 Nov 2015, 20:57
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Re: Series A(n) is such that i*A(i) = j*A(j) for any pair of [#permalink]
27 Nov 2015, 23:22
Bunuel wrote:
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.
I don't understand this part. How could I know that A(1) = 1 The question stem mentioned only that "A(1) is a positive integer"
If A(1) = 2, then ---> 1*A(1) = 2*A(2) ---> A(2) = 1 Then II cannot be true.
Re: Series A(n) is such that i*A(i) = j*A(j) for any pair of [#permalink]
28 Nov 2015, 07:29
Expert's post
pakasaip wrote:
Bunuel wrote:
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.
I don't understand this part. How could I know that A(1) = 1 The question stem mentioned only that "A(1) is a positive integer"
If A(1) = 2, then ---> 1*A(1) = 2*A(2) ---> A(2) = 1 Then II cannot be true.
Please tell me if I get something wrong.
Thanks
Please notice that it says "IF \(a_1=1\), ..." and also that the question asks which of the following is possible, so which of the following could be true. _________________
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