Problem 4 - 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. $$ 1,5,25,125,625, \ldots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio \(r = 5\).

Step 1: Identify if the Sequence is Geometric

A sequence is geometric if each term after the first is found by multiplying the previous one by a constant number (called the common ratio, \(r\)). Start by examining the given sequence: \(1, 5, 25, 125, 625, \ldots\). To check if it is geometric, calculate the ratio between consecutive terms.

Step 2: Calculate the Consecutive Ratios

Find the ratio of the second term to the first term: \( \frac{5}{1} = 5 \). Then, find the ratio of the third term to the second term: \( \frac{25}{5} = 5 \). Continue this for the fourth to the third \( \frac{125}{25} = 5 \) and the fifth to the fourth \( \frac{625}{125} = 5 \).

Step 3: Verify Consistency of the Ratios

Since the ratio between each consecutive term is consistently \(5\), it indicates the sequence is geometric. All ratios are equal, confirming a common ratio \(r = 5\).

Step 4: Conclusion

The sequence is geometric with a common ratio \( r = 5 \). All terms follow the rule of multiplying the previous term by \(5\).

Key Concepts

Understanding the Common Ratio in Geometric Sequences

The common ratio is an integral part of a geometric sequence. It is the factor by which each term in a sequence is multiplied to get the next term. When examining a sequence like \(1, 5, 25, 125, 625, \ldots\), our goal is to determine if there is such a fixed number that transforms each term to the next.

To find this common ratio \(r\), we look at the ratio of every pair of consecutive terms:
\[- \text{For } 5, \text{the first ratio is } \frac{5}{1} = 5,\]
\[- \text{then } \frac{25}{5} = 5,\]
\[- \frac{125}{25} = 5, \text{ and }\frac{625}{125} = 5.\]
This consistency tells us the sequence is geometric with a common ratio of \(r = 5\). Every time, the product of multiplying the last term by \(5\) gives us the next term.

Understanding the common ratio helps us not only identify but also predict further terms in the sequence.

Sequence Verification Process

When faced with any sequence, the verification process is essential to confirm whether it is geometric. Verification involves checking the ratios between terms to see if they remain constant.

Here is what to do:

• Start with the first term and calculate the ratio of each subsequent term to the previous one.
• Ensure each calculated ratio is identical. If one or more ratios differ, the sequence is not geometric.

This method can be applied to any proposed sequence.

If all calculated ratios match, we conclude the sequence to be geometric. This tells us there is a repetitive pattern governed by the common ratio, reinforcing our understanding of how each term relates to the previous one.

Performing a Consistency Check

Consistency checks serve as the final confirmation in determining whether a sequence is geometric. Even after identifying a common ratio, it's crucial to ensure every calculated ratio between terms is consistent.

Consider our sequence again: \(1, 5, 25, 125, 625, \ldots\). Each calculation shows the consistent ratio of \(5\), confirming that this pattern holds true throughout.

• Check each pair: \( \frac{5}{1}, \frac{25}{5}, \frac{125}{25}, \frac{625}{125} \).
• All ratios equal to \(5\) prove the sequence is consistent.

Once we've verified consistency, we are assured of the sequence's geometric nature.

This process underscores the importance of verifying the consistency of the identified common ratio across all terms to establish the geometric property of any sequence fully.