Problem 13 - Algebra and Trigonometry

Question

Determine whether the sequence is geometric. If it is geometric, find the common ratio. $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$

Step-by-Step Solution

Answer

The sequence is not geometric, as the ratios between consecutive terms are not constant.

Step 1: Understand the Definition

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. We need to check if such a relationship exists between consecutive terms in the given sequence.

Step 2: Identify Consecutive Terms

First, identify pairs of consecutive terms. For example, the first pair is \( \frac{1}{2} \) and \( \frac{1}{3} \), the second pair is \( \frac{1}{3} \) and \( \frac{1}{4} \), and so on.

Step 3: Calculate Ratios

Calculate the ratio of each consecutive pair of terms by dividing the second term by the first. For the first pair: \[ \text{Ratio} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3} \]For the second pair: \[ \text{Ratio} = \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{1}{4} \times \frac{3}{1} = \frac{3}{4} \]

Step 4: Compare Ratios

Compare the calculated ratios. If the sequence is geometric, all ratios must be equal. So far, we have the ratios \( \frac{2}{3} \) and \( \frac{3}{4} \). These ratios are not equal.

Step 5: Conclusion

Since the ratios between consecutive terms are not constant, the sequence is not geometric.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key characteristic. It refers to the constant factor you multiply by to move from one term to the next. Determining the common ratio involves selecting any two consecutive terms and performing a division where the second term is divided by the first.

This constant multiplier transforms an arithmetic sequence to a geometric one. Here's an easy way to conceptualize it:

• Take any two consecutive terms: let's say, the terms are represented as \( a_1 \) and \( a_2 \).
• Calculate the ratio \( r \) using the formula \( r = \frac{a_2}{a_1} \).
• Check if this value remains consistent throughout the sequence. If it does, you have identified the common ratio, confirming the sequence is geometric.

Unfortunately, in the given sequence \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \), this ratio was not consistent (as shown by different values of \( \frac{2}{3} \) and \( \frac{3}{4} \)), confirming that the sequence isn't geometric.

Consecutive Terms

Understanding consecutive terms is crucial to evaluating sequences. Consecutive terms are simply terms that appear one after the other in a sequence. In the given sequence, these are pairs like \( \frac{1}{2} \) and \( \frac{1}{3} \).In analysis, we look at each consecutive pair to determine how the sequence progresses. Specifically:

• Identify two consecutive terms.
• Calculate what happens mathematically as you move from one to the next. This involves multiplication by the common ratio in a geometric sequence.

In our example, examining consecutive terms highlights a changing ratio, and since the relationship isn't constant, the sequence is not geometric.

Sequence of Numbers

A sequence of numbers is an ordered collection of numbers defined by a specific rule. Sequences can vary widely in their structures, such as arithmetic or geometric, each having a distinct method of progression.In a geometric sequence, each term is derived by multiplying the previous one by a constant common ratio. This repeatable operation provides predictable and consistent term generation. To recognize a sequence:

• Look for a repeating pattern or rule, such as constant multiplication in geometric sequences.
• Check if every term adheres to the defined pattern.

In the sequence \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \) despite being orderly, the lack of a constant multiplical relationship derails it from fitting into the geometric sequence framework, underscoring the importance of distinguishing the type of sequence clearly.