In sequence of 9 distinct numbers {a1, a2, a3, ..., a9}, nth term is g

Question

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

BrentGMATPrepNow · Answer

Great question Bunuel !

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

BRLLK0800 · Answer

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

Bunuel · Answer

b cannot be 0, because in this case  a_n = a_{n−1} , which would make all terms equal. But we are told that the numbers in the sequence are  distinct , so b must be nonzero.

bumpbot · Answer

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In sequence of 9 distinct numbers {a1, a2, a3, ..., a9}, nth term is g

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In sequence of 9 distinct numbers {a1, a2, a3, ..., a9}, nth term is g

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