ardz24 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
Statement 1: a₉ = 81
Case 1: k=81
a₈ = a₉/k = 81/81 = 1.
a₇ = a₈/k = 1/81.
a₆, a₅, a₄, a₃, a₂, and a� will all be less than 1/81 and thus less than 1.
Here, the first 7 terms are all less than 1, for a total of 7 terms less than 1.
Case 2: k=9
a₈ = a₉/k = 81/9 = 9.
a₇ = a₈/k = 9/9 = 1.
a₆ = a₇/k = 1/9.
a₅, a₄, a₃, a₂, and a� will be less than 1/9 and thus less than 1.
Here, the first 6 terms are all less than 1, for a total of 6 terms less than 1.
Since the number of terms less than 1 can be different values, INSUFFICIENT.
Statement 2: aâ‚… = 1
Case 3: k=2
a₆ = ka₅ = 2*1 = 2
aâ‚„ = aâ‚…/k = 1/2.
a₇, a₈ and a₉ will all be greater than 2 and thus greater than 1.
a₃, a₂, and a� will all be less than 1/2 and thus less than 1.
Here, the first 4 terms are all less than 1, for a total of 4 terms less than 1.
Case 4: k=1/2
a₆ = ka₅ = (1/2)(1) = 1/2.
aâ‚„ = aâ‚…/k = 1/(1/2) = 2.
a₇, a₈ and a₉ will all be less than 1/2 and thus less than 1.
a₃, a₂, and a� will all be greater than 2 and thus greater than 1.
Here, the last 4 terms are all less than 1, for a total of 4 terms less than 1.
Cases 3 and 4 illustrate that -- whether k is a positive integer or a positive fraction -- there will be a total of 4 terms less than 1.
SUFFICIENT.
The correct answer is
B.
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