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The sequence of non-zero numbers \(a_1\), \(a_2\), \(a_3\), ... is such that \(a_{n} = \frac{a_{n-2}} {a_{n-1}}\) for all \(n ≥ 3\). Is \(a_9 > 0\)?
(1) \(a_3 < 0\)
(2) \(a_1*a_2 < 0\)
(1) \(a_3 < 0\)
(2) \(a_1*a_2 < 0\)
30 Dec 2023, 23:31
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Official Solution:
The sequence of non-zero numbers \(a_1\), \(a_2\), \(a_3\), ... is such that \(a_{n} = \frac{a_{n-2}} {a_{n-1}}\) for all \(n ≥ 3\). Is \(a_9 > 0\)?
(1) \(a_3 < 0\)
This implies that \(a_1\) and \(a_2\) have different signs: either one is positive and the other is negative, or vice versa. Let's examine both scenarios:
If \(a_1\) is positive and \(a_2\) is negative, then we'd have the following pattern:
+ - - + - - + - -
So, \(a_9 < 0\).
If \(a_1\) is negative and \(a_2\) is positive, then we'd have this pattern:
- + - - + - - + -
In this case, \(a_9 < 0\) as well.
Sufficient.
(2) \(a_1*a_2 < 0\)
This statement essentially provides the same information as the first: \(a_1\) and \(a_2\) have different signs. As we have established, this is sufficient to answer the question.
Answer: D
The sequence of non-zero numbers \(a_1\), \(a_2\), \(a_3\), ... is such that \(a_{n} = \frac{a_{n-2}} {a_{n-1}}\) for all \(n ≥ 3\). Is \(a_9 > 0\)?
(1) \(a_3 < 0\)
This implies that \(a_1\) and \(a_2\) have different signs: either one is positive and the other is negative, or vice versa. Let's examine both scenarios:
If \(a_1\) is positive and \(a_2\) is negative, then we'd have the following pattern:
+ - - + - - + - -
So, \(a_9 < 0\).
If \(a_1\) is negative and \(a_2\) is positive, then we'd have this pattern:
- + - - + - - + -
In this case, \(a_9 < 0\) as well.
Sufficient.
(2) \(a_1*a_2 < 0\)
This statement essentially provides the same information as the first: \(a_1\) and \(a_2\) have different signs. As we have established, this is sufficient to answer the question.
Answer: D
