Problem 101 - College Algebra

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

The statement makes sense. By applying different non-zero ratios, an infinite number of distinct geometric sequences starting with 5 can be generated. This is due to the properties of geometric sequences where a variation in ratio produces different sequences.

Step 1: Understand the statement

First, ensure an understanding of the provided statement. Summarized, it states that nonzero ratios \(r\) can be selected and 5 (the first term) can be multiplied by each value of \(r\) repeatedly to create an infinite number of geometric sequences.

Step 2: Analyze the logic in the statement

In a geometric sequence, it's true that changing the ratio (the number by which each term is multiplied to get the next term) will yield a different sequence, even if the first term remains the same. Given the infinite possibilities for the \(r\) value in the set of nonzero numbers, indeed, an infinite number of geometric sequences can be generated.

Step 3: Validate the statement

Consider this: a geometric sequence starting with 5 and ratio 2 will be {5, 10, 20, 40...}. If the ratio is changed to 3, the sequence will be {5, 15, 45, 135...}. The sequences are distinct, affirming the validity of the statement in the question.

Step 4: Final verification

Because the ratio can be any non-zero number (positive, negative, fractions, decimals, irrationals, etc.), it means there are infinite possibilities for \(r\). Considering this, it can indeed be concluded that an infinite number of geometric sequences can be generated, all commencing with the first term as 5.