BTGmoderatorDC 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
OA B
Source: Manhattan Prep
Let's take each statement one by one.
(1) The ninth term in this sequence is 81.
Case 1: Say k = 3, then the first nine terms are 1/81, 1/27, 1/9, 1/3, 1, 3, 9, 27, and 81. The terms greater than 1 are 3, 9, 27, and 81. The answer is 4.
Case 2: Say k = 9, then the first nine terms are ..., 1/81, 1/9, 1, 9, and 81. The terms greater than 1 are 9 and 81. The answer is 2.
No unique answer. Insufficient.
(2) The fifth term in this sequence is 1.
Case 1: Say k = 3. The nine terms are 1/81, 1/27, 1/9, 1/3, 1, 3, 9, 27, and 81. The terms greater than 1 are 3, 9, 27, and 81. The answer is 4.
Case 2: Say k = 1/3. The nine terms are 81, 27, 9, 3, 1, 1/3, 1/9, 1/27, and 1/81. The terms greater than 1 are 81, 27, 9, and 3. The answer is still 4.
Unique answer. Sufficient.
In another way, since the 5th term = 1 and k ≠1, either the 1st, 2nd, 3rd, and 4th terms would be greater than 1 OR the 6th, 7th, 8th, and 9th terms would be greater than 1. In either case, the number of terms greater than 1 is 4.
The correct answer:
B
Hope this helps!
-Jay
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