<p>The sixth term in a certain geometric sequence y=ab^x is greater than both the fifth term and the seventh term in this sequence.Which of the following must be ture?</p>
<p>I. The tenth term is greater than the third term.</p>
<p>II. The sixth term is greater than the fourth term.</p>
<p>III. The product of any two consecutive terms is less than zero.</p>
<p>(a)None
(b)I only
(c)II only
(d)I and III
(e) I,II,and III</p>
<p>First, realize that the common ratio b is negative (otherwise, the geometric sequence is strictly increasing or strictly decreasing, or constant). So the sequence alternates +, -, +, -. Since the sixth term is greater than the 5th and 7th terms, all even-positioned terms are positive and all odd-positioned terms are negative.</p>
<p>I: I is true because the 10th term is positive and the 3rd term is negative.</p>
<p>II: Does not have to be true. Both are positive, but the common ratio b could be between 0 and -1. If so, the 6th term is smaller than the 4th term.</p>
<p>III: This is true since consecutive terms alternate sign.</p>
<p>The answer is (d) I and III.</p>