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27 May 2022, 03:32
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B
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56%
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In sequence of 9 distinct numbers \(\{ a_1, \ a_2, \ a_3, \ ..., \ a_9 \} \), nth term is given by \(a_n = a_{n−1} + b\), where \(2 ≤ n ≤ 9\) and b is a constant. How many of the terms in the sequence are negative?
(1) \(a_1 = 16\)
(2) \(a_5 = 0\)
(1) \(a_1 = 16\)
(2) \(a_5 = 0\)
27 May 2022, 12:33
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Bunuel
In sequence of 9 distinct numbers \(\{ a_1, \ a_2, \ a_3, \ ..., \ a_9 \} \), nth term is given by \(a_n = a_{n−1} + b\), where \(2 ≤ n ≤ 9\) and b is a constant. How many of the terms in the sequence are negative?
(1) \(a_1 = 16\)
(2) \(a_5 = 0\)
Great questionBunuel!(1) \(a_1 = 16\)
(2) \(a_5 = 0\)
Given: In sequence of 9 distinct numbers \(\{ a_1, \ a_2, \ a_3, \ ..., \ a_9 \} \), nth term is given by \(a_n = a_{n−1} + b\), where \(2 ≤ n ≤ 9\) and b is a constant.
Important: This information tells us that to find any term after the first term, we add \(b\) to the previous term.
So, if b is positive, then each successive term will be greater than the term before it.
Conversely, if b is negative, then each successive term will be less than the term before it.
Note: If b is 0, then all of the terms will be the same, BUT we're told the sequence has 9 distinct numbers. So, b can't equal 0.
Target question: How many of the terms in the sequence are negative?
Statement 1: \(a_1 = 16\)
There are several different sequences that satisfy statement 1. Here are two:
Case a: If b = -5, then the first 9 terms of the sequence are 16, 11, 6, 1, -4, -9, -14, -19, -24. In this case, the answer to the target question is there are 5 negative terms in the sequence
Case b: If b = 1, then the first 9 terms of the sequence are 16, 17, 18, 19, 20, 21, 22, 23, 24. In this case, the answer to the target question is there are 0 negative terms in the sequence
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(a_5 = 0\)
Keep in mind that EITHER the sequence increases with each successive term, OR the sequence decreases with each successive term.
Since \(a_5\) is the middle term of the first 9 terms of the sequence, we can be certain that exactly 4 of the first 9 terms of the sequence will be negative and 4 will be positive.
Consider these two cases:
If b = -2, then the first 9 terms of the sequence are 8, 6, 4, 2, 0, -2, -4, -6, -8. In this case, the answer to the target question is there are 4 negative terms in the sequence
If b = 1, then the first 9 terms of the sequence are -4, -3, -2, -1, 0, 1, 2, 3, 4. In this case, the answer to the target question is there are 4 negative terms in the sequence
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer: B
LucasCredidio
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27 Jun 2025, 20:48
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What if b=0? Then the number of negative would be different, saying that b is a constant doesnt mean it can ́t be zero
