Problem 16 - Algebra and Trigonometry

Question

Determine whether the sequence is geometric. If it is geometric, find the common ratio. $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \dots $$

Step-by-Step Solution

Answer

The sequence is not geometric; there is no common ratio.

Step 1: Check for a common ratio

To determine if the sequence is geometric, we need to check if there is a common ratio between consecutive terms. Divide the second term by the first term: \( \frac{1}{4} \div \frac{1}{2} = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} \).

Step 2: Check the next pair of terms

Apply the same process to the next pair of terms. Divide the third term by the second: \( \frac{1}{6} \div \frac{1}{4} = \frac{1}{6} \times \frac{4}{1} = \frac{4}{6} = \frac{2}{3} \).

Step 3: Compare the ratios

For a sequence to be geometric, the ratio must be the same between all consecutive terms. The first calculated ratio was \( \frac{1}{2} \) and the second was \( \frac{2}{3} \), which are not the same.

Key Concepts

Common Ratio

In a geometric sequence, one of the most critical elements is the common ratio. This ratio is what you multiply by each term to get the next term in the sequence. Understanding the common ratio is essential for identifying and working with geometric sequences.
To find the common ratio, you divide any term in the sequence by the previous term. For example, in the sequence analyzed, which starts as \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots \), you would attempt to find the ratio between the first term (\( \frac{1}{2} \)) and the second term (\( \frac{1}{4} \)) by computing \( \frac{1}{4} \div \frac{1}{2} = \frac{1}{2} \).

• If this ratio remains constant as you move through the sequence, then you have identified the common ratio, confirming that the sequence is geometric.
• If it does not, no common ratio exists, and the sequence is not geometric.

Sequence Analysis

Sequence analysis refers to the process of examining a sequence to understand its nature, such as whether it's arithmetic, geometric, or neither. In this exercise, sequence analysis involves scrutinizing the numbers to determine if a geometric sequence is present.
To analyze a sequence for geometric characteristics, one checks for a constant common ratio. You need to calculate the ratio between consecutive terms multiple times:

• First, from \( \frac{1}{2} \) to \( \frac{1}{4} \), the ratio is \( \frac{1}{2} \).
• Next, from \( \frac{1}{4} \) to \( \frac{1}{6} \), the ratio changes to \( \frac{2}{3} \).

Analyzing this shows the sequence is not geometric as these ratios are different. Such analysis provides clarity on the absence of a constant pattern or rule.

Non-Geometric Sequence

A non-geometric sequence is one where there is no consistent rule for multiplying each term to get the next, which is the opposite of what defines a geometric sequence. This occurs when the calculated ratios between pairs of terms vary.
In the given sequence \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots \), the lack of a single, repeating common ratio confirms it's non-geometric. Different ratios were found when comparing successive terms:

• \( \frac{1}{4} \div \frac{1}{2} = \frac{1}{2} \)
• \( \frac{1}{6} \div \frac{1}{4} = \frac{2}{3} \)

Since \( \frac{1}{2} eq \frac{2}{3} \), the sequence does not fit the criteria for a geometric pattern. Understanding non-geometric sequences helps in recognizing diverse forms of sequences beyond just constant ratio-based sequences, aiding in broader mathematical comprehension.