Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ \frac{1}{2}, 2,8,32, \ldots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 4.

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, denoted as \(r\).

Step 2: Calculate the Ratios

To determine if the sequence is geometric, calculate the ratio between consecutive terms in the sequence. - First ratio: \( \frac{2}{\frac{1}{2}} = 2 \times \frac{1}{1/2} = 4 \)- Second ratio: \( \frac{8}{2} = 4 \)- Third ratio: \( \frac{32}{8} = 4 \)

Step 3: Analyze the Ratios

Since all the calculated ratios are equal, we recognize that the sequence has a consistent common ratio of \(4\). This confirms the sequence is geometric.

Step 4: Conclude

Since we have determined the ratios are consistent across the terms of the sequence, the sequence \( \frac{1}{2}, 2, 8, 32, \ldots \) is geometric with a common ratio \(r = 4\).

Key Concepts

Common Ratio

In any geometric sequence, a common ratio is the key element that defines the nature of the sequence. It's the constant factor by which each term is multiplied to obtain the next term in the sequence. For example, in the sequence \( \frac{1}{2}, 2, 8, 32, \ldots \), we determine if it's geometric by evaluating whether each term can be consistently obtained by multiplying the previous term using a fixed number. If we discover during our calculations that the ratio \( r \), calculated for each pair of successive terms, remains the same, we found our common ratio. This is what makes the sequence not only geometric but predictable in its progression.

Sequence Analysis

To decide if a sequence is geometric, sequence analysis is crucial. It involves examining the progression of terms and checking for a consistent pattern. With the sequence \( \frac{1}{2}, 2, 8, 32, \ldots \), we start by comparing each term with the one before it. Sequence analysis means taking a step-by-step approach and doing the math needed to spot trends that define the sequence type.

• Identify the initial terms of the sequence.
• Calculate the ratio between each pair of consecutive terms.
• Check if these ratios are consistent throughout.

If together these analyses show uniformity, you likely have a geometric sequence, establishing a foundation upon which all future terms in the sequence can be predicted confidently.

Ratio Calculation

Determining if a sequence like \( \frac{1}{2}, 2, 8, 32, \ldots \) is geometric involves precise ratio calculation between consecutive terms. Here's how you can do it:
- Begin with calculating the first ratio, \( \frac{2}{\frac{1}{2}} \). This simplifies to \(4\) as multiplying by the reciprocal, \( 2 \times \left( \frac{1}{1/2} \right) \), gives \(4\).
- Continue the calculation process with the second ratio: \( \frac{8}{2} \), which equals \(4\).
- Finally, evaluate the third ratio: \( \frac{32}{8} \), which also results in \( 4 \).
Noticing that the results are all consistent, we affirm that the common ratio is \( 4 \), proving the sequence is geometric. Accurate ratio calculation solidifies understanding and confidently confirms the sequence's geometric nature.