Problem 20 - Precalculus Mathematics for Calculus

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.

Step 1: Identify common ratios

First, we find the ratio between each pair of consecutive terms in the sequence to check if they are equal. For the first two terms: \( r_1 = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \). For the next two terms: \( r_2 = \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3} \). Check the next terms: \( r_3 = \frac{\frac{1}{8}}{\frac{1}{6}} = \frac{3}{4} \).

Step 2: Determine whether the sequence has a common ratio

Observe that the found ratios are \( \frac{1}{2} \), \( \frac{2}{3} \), and \( \frac{3}{4} \), which are different from each other. A geometric sequence must have the same common ratio for all consecutive terms.

Step 3: Conclusion: Determine type of sequence

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

Key Concepts

Sequence Identification

Identifying the type of sequence is the first step in understanding any series of numbers. In mathematics, when given a sequence, we determine if it fits a particular classification. This involves looking for patterns or rules that apply to the sequence.
Most common types include arithmetic sequences and geometric sequences.

• An arithmetic sequence has a constant difference between consecutive terms.
• A geometric sequence has a constant ratio between consecutive terms.

In the provided exercise, the sequence is given as \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots \).
To identify it, we checked for a consistent pattern or ratio. Since this sequence does not have a common ratio between every pair of terms, it is not geometric.

Common Ratio

The common ratio is a key characteristic of a geometric sequence. It’s the factor by which we multiply one term to get to the next term in the sequence.
Mathematically, for a sequence \( a_1, a_2, a_3, \ldots \), the common ratio \( r \) is determined by \( r = \frac{a_{n+1}}{a_n} \) for all terms \( n \).
To confirm a geometric sequence, every consecutive term must share the same common ratio.
In the exercise, we attempted to find this ratio using specific pairs:

• Between \( \frac{1}{2} \) and \( \frac{1}{4} \), the ratio is \( \frac{1}{2} \).
• Between \( \frac{1}{4} \) and \( \frac{1}{6} \), the ratio is \( \frac{2}{3} \).
• Between \( \frac{1}{6} \) and \( \frac{1}{8} \), the ratio is \( \frac{3}{4} \).

The differing ratios indicate the absence of a common ratio, confirming that the sequence isn't geometric.

Consecutive Terms

Consecutive terms are simply terms that come one after another in a sequence. When evaluating sequences, we often look at these successive terms to determine the nature of the sequence.
For our sequence \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots \), each term is the direct successor of the preceding one.
Whether a sequence is arithmetic, geometric, or anything else often depends on the relationships between these consecutive terms.
In geometric sequences, multiplying a term by the common ratio yields the next term. But, as we found with our sequence, the step-wise differences by multiplication were inconsistent.
Therefore, analyzing these consecutive terms helped confirm the sequence isn't geometric due to the lack of a consistent relationship.

Non-Geometric Sequence

A non-geometric sequence is any sequence that doesn't have a consistent common ratio between consecutive terms. This term broadly applies to sequences that don't fit the strict definition of geometric.
Our examined sequence \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots \) didn't exhibit a constant ratio. Therefore, it serves as an example of a non-geometric sequence.
Understanding non-geometric sequences is essential in determining their unique characteristics or patterns.

• They could be arithmetic, where differences are consistent.
• They might have no regular pattern, making them neither arithmetic nor geometric.

Recognizing a sequence as non-geometric is the first step in exploring these other possible patterns or rules that govern it.