BTGmoderatorLU wrote:Source: Princeton Review

Sequence S consists of 24 nonzero integers. If each term in S after the second is the product of the previous two terms, how many terms in S are negative?

1) The third term in S is positive.
2) The fourth term in S is negative.

The OA is B

Let's take each statement one by one.

1) The third term in S is positive.

Say the first three terms are x, y and z. We know that z is positive; thus, x and y both must have same signs: either both positive or both negative. The count of negative terms in S cannot be uniquely determined. Insufficient.

2) The fourth term in S is negative.

Say the first three terms are x, y, z and p. We know that p is negative; thus, y and z must have opposite signs:

Case 1: Say y is positive and z is negative.

=> The four terms are x, |y|, -|z|, -|p|. Since the third term is negative and the second term is positive, the first term must be negative.

Thus, the four terms are -|x|, |y|, -|z|, -|p|. So, we have uniquely determined the signs of the first four terms: three negatives and one positive; thus, we can determine the signs of the remaining 20 terms. The 5th, 7th, 9th, ..., 23rd term would be positive (10 terms) and 6th, 8th, 10th, ... and 24th term would be negative (10 terms). So, there are 13 negative terms.

Case 2: Say y is negative and z is positive.

=> The four terms are x, -|y|, |z|, -|p|. Since the third term is positive and the second term is negative, the first term must be negative.

Thus, the four terms are -|x|, -|y|, |z|, -|p|. So, we have uniquely determined the signs of the first four terms: three negatives and one positive; thus, we can determine the signs of the remaining 20 terms. The 5th, 7th, 9th, ..., 23rd term would be negative (10 terms) and 6th, 8th, 10th, ... and 24th term would be positive (10 terms). So, there are 13 negative terms. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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