In a geometric sequence where the first term is positive and [tex]$r\ \textgreater \ 1$[/tex], the terms always increase.
A. True
B. False
Answer (1)
A geometric sequence is defined by a first term a 1 and a common ratio r .
Given 0"> a 1 > 0 and 1"> r > 1 , the terms of the sequence are analyzed.
As n increases, r n − 1 increases because 1"> r > 1 .
Since a 1 is positive, the terms a n = a 1 n − 1 always increase. Therefore, the statement is True .
Explanation
Problem Analysis Let's analyze the given information. We have a geometric sequence with a positive first term ( 0"> a 1 > 0 ) and a common ratio 1"> r > 1 . We need to determine if the terms of the sequence always increase.
General Term The general term of a geometric sequence is given by a n = a 1 ⋅ r n − 1 , where a 1 is the first term, r is the common ratio, and n is the term number.
Analyzing the terms Since 0"> a 1 > 0 and 1"> r > 1 , we can analyze how a n changes as n increases. As n increases, n − 1 also increases. Since 1"> r > 1 , r n − 1 will increase as n − 1 increases.
Conclusion Since a 1 is positive, multiplying an increasing value r n − 1 by a 1 will result in an increasing sequence. Therefore, the terms of the geometric sequence will always increase.
Examples
Geometric sequences are used in various real-world applications, such as calculating compound interest, modeling population growth, and determining the depreciation of assets. For example, if you invest $1000 in an account that earns 5% interest compounded annually, the amounts at the end of each year form a geometric sequence with a first term of $1000 and a common ratio of 1.05. Understanding geometric sequences helps you predict how your investment will grow over time.
