Needgmat wrote:The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠1. How many of the first nine terms in this sequence are greater than 1?
(1) The ninth term in this sequence is 81.
(2) The fifth term in this sequence is 1.
We are given that the next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠1, and we need to determine how many of the first nine terms in this sequence are greater than 1.
Statement One Alone:
The ninth term in this sequence is 81.
Without knowing any other terms in the sequence or the value of the positive constant k we are multiplying, we can't determine the number of terms in the sequence that are greater than 1. For example, if k = 3, then by going backward we have:
9th term = 81, 8th term = 27, 7th term = 9, 6th term = 3, 5th term = 1, 4th term = 1/3, and so on.
In this case, we have 4 terms that are greater than 1.
However, if k = 9, then by going backward again we have:
9th term = 81, 8th term = 9, 7th term = 1, 6th term = 1/9, and so on.
In this case we have only 2 terms that are greater than 1. Therefore, statement one alone is not sufficient. We can eliminate answer choices A and D.
Statement Two Alone:
The fifth term in this sequence is 1.
Since there are 9 total terms, we see that the 5th term is the middle term in our sequence. In other words there are 4 terms below the 5th term and 4 terms above the 5th term. Since we know that k is positive, it's either a positive proper fraction or a number greater than 1. Thus, regardless of whether k is a positive proper fraction or a number greater than 1, there will be 4 numbers above 1 and 4 numbers below 1.
For instance, if k = 1/2, then the 1st to the 4th terms, inclusive, are greater than 1, and if k = 2, then the 6th to the 9th terms, inclusive, are greater than 1. Either way, there are 4 terms greater than 1. Statement two alone is sufficient to answer the question.
Answer:
B
