Problem 29 - Contemporary Precalculus

Question

In Exercises \(23-30,\) show that the given sequence is geometric and find the common ratio. $$\left\\{10 e^{4 n}\right\\}$$

Step-by-Step Solution

Answer

Question: Determine if the given sequence \(\{10 e^{4n}\}\) is geometric, and if so, find the common ratio. Answer: The sequence \(\{10 e^{4n}\}\) is geometric with a common ratio of \(e^4\).

Step 1: Identify the sequence and the general term

The given sequence is \(\{10 e^{4n}\}\). The general term in the sequence is of the form \(a_n = 10 e^{4n}\) where \(n = 0, 1, 2, \dots\). Step 2: Calculate the ratio between consecutive terms

Step 2: Calculate the ratio between consecutive terms

To find the ratio between consecutive terms, divide a term by the previous term. We can find the ratio: \(r = \dfrac{a_{n + 1}}{a_n} = \dfrac{10e^{4(n + 1)}}{10e^{4n}}\) Step 3: Simplify the ratio

Step 3: Simplify the ratio

To simplify the ratio, we can apply exponent rules: \(r = \dfrac{10e^{4(n + 1)}}{10e^{4n}} = \dfrac{10e^{4n+4}}{10e^{4n}} = \dfrac{e^{4n}e^{4}}{e^{4n}} = e^4\) Step 4: Conclusion

Step 4: Conclusion

The ratio \(r\) is a constant, which means the given sequence \(\{10 e^{4n}\}\) is a geometric sequence. The common ratio for this sequence is \(r = e^4\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is very important. It's the constant factor by which we multiply one term to get to the next term in the sequence.
To determine the common ratio, we divide any term by the previous term. This is precisely what was performed in the solution provided: the ratio

• Given the sequence \[\{10 e^{4n}\}\], the terms can be written as \[10 e^{4 \cdot 0}, 10 e^{4 \cdot 1}, 10 e^{4 \cdot 2}, \dots\]
• Calculating the ratio between the terms \[\dfrac{10 e^{4(n+1)}}{10 e^{4n}}\] results in \[e^4\].

The common ratio of \[e^4\] shows that each term is \[e^4\] times its preceding term. This commonality confirms the sequence is indeed geometric. Simple tests of subsequent numbers will consistently return the same ratio, solidifying this concept.

Exponent Rules

Understanding exponent rules will make simplifying expressions in a geometric sequence much easier. Here we deal with powers and how to handle them properly.
Exponent rules provide a systematic approach to manage complex expressions:

• Product Rule: When multiplying powers with the same base, add the exponents. \[a^m \cdot a^n = a^{m+n}\]
• Quotient Rule: When dividing powers with the same base, subtract the exponents. \[\dfrac{a^m}{a^n} = a^{m-n}\]
• Power of Power Rule: When raising a power to another power, multiply the exponents. \[(a^m)^n = a^{mn}\]
• Being savvy with these rules helps in reducing complex expressions quickly and correctly.

In the example provided, using exponent rules resulted in simplifying \[\dfrac{e^{4n+4}}{e^{4n}}\] to \[e^4\]. This step was crucial to identify the common ratio correctly.

Sequence Identification

Identifying what kind of sequence you’re looking at is vital before solving any problems associated with it. Distinguishing between an arithmetic and a geometric sequence can guide your approach.
To identify a sequence neatly as geometric, ask if a constant multiplier exists between consecutive terms:

• Geometric Sequence: Multiplier remains constant. For instance: In the sequence \[\{10 e^{4n}\}\]
• The multiplier between consecutive terms is confirmed to be the common ratio \[e^4\].
• This constant multiplier is the exact identifier that tells you it is a geometric sequence.
• Look for an exponential pattern. Here, the sequence’s terms are clearly exponential expressions, fitting the geometric profile.

Using these insightful checks and the calculations demonstrated, you efficiently identify whether the sequence is geometric or not. This skill is foundational for tackling both theoretical and practical mathematical challenges.