Problem 101 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Step-by-Step Solution

Answer

Yes, the statement makes sense. As there are infinite possible nonzero values for \(r\), with each unique value, a new geometric sequence starting with 5 can be formed. So, an infinite number of geometric sequences can be generated.

Step 1: Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence can be written as \(a, ar, ar^2, ar^3, ...\), where \(a\) is the first term and \(r\) is the common ratio.

Step 2: Analyzing the Statement

The statement provided says that an infinite number of geometric sequences can be generated with the first term as 5 by picking different nonzero values of \(r\) and multiplying 5 by each value of \(r\) repeatedly. This means, for each unique value of \(r\), a different geometric sequence is obtained.

Step 3: Reasoning and Conclusion

As there are infinite possible nonzero values of \(r\) (both positive and negative), with every unique value of \(r\), a new geometric sequence can be generated. Therefore, the statement makes sense and is correct.