Problem 18 - Precalculus Mathematics for Calculus

Question

Determine whether the sequence is geometric. If it is geometric, find the common ratio. $$e^{2}, e^{4}, e^{6}, e^{8}, \ldots$$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(e^2\).

Step 1: Identifying the Sequence

Look at the sequence given: \(e^2, e^4, e^6, e^8, \ldots\). Each term in the sequence can be represented as \(e^{2n}\) where \(n\) is an integer starting from 1.

Step 2: Understanding the Geometric Sequence Definition

A sequence is considered geometric if the ratio of any term to its preceding term is constant.

Step 3: Finding Common Ratios

Calculate the ratio between successive terms: \(\frac{e^4}{e^2} = e^2\), \(\frac{e^6}{e^4} = e^2\), and \(\frac{e^8}{e^6} = e^2\). Show that each ratio is equal.

Step 4: Conclusion

Since the ratio between each consecutive term is the same \(e^2\), the sequence is indeed geometric with a common ratio of \(e^2\).

Key Concepts

Common Ratio

In a geometric sequence, the term 'common ratio' holds great significance. It is the factor by which we multiply each term to obtain the subsequent term in the sequence. Calculating the common ratio involves examining consecutive terms of the sequence.
For example, if you have a sequence like \( e^2, e^4, e^6, e^8, \ldots \), you observe the ratio between two successive terms to determine constancy. So, you compute \( \frac{e^4}{e^2} = e^2 \), then similarly, \( \frac{e^6}{e^4} = e^2 \). If these quotients are the same, you have found your common ratio.
Remember, a geometric sequence retains its nature only when this constant multiplier, the common ratio, is present, ensuring the sequence's smooth progression. Thus, recognizing the common ratio is crucial for identifying and understanding geometric sequences.

Exponential Function

The nature of the sequence \( e^2, e^4, e^6, e^8, \ldots \) is grounded in exponential functions. An exponential function can generally be represented in the form \( a^n \), where \( a \) is a constant and \( n \) is an exponent (often a variable or integer).
Specifically, the sequence seen here can be expressed as \( e^{2n} \). Each exponentiation of \( e \) results in values that differ by consistent multiplicative factors, illustrating both the power and growth behavior of exponential functions.
Exponential functions are vital in various fields like biology, finance, and physics because they model situations involving constant relative growth rates, much like the uniform ratio in a geometric sequence. Understanding the link between exponential functions and geometric sequences offers insights into how these sequences expand and evolve.

Sequence Identification

To determine whether a sequence is geometric, you'll need to understand sequence identification. This process involves analyzing how one term changes to the next to establish the sequence type.
Begin by examining whether there is a consistent pattern. For our sequence \( e^2, e^4, e^6, e^8, \ldots \), notice that each term increases in the exponent by 2. Consequently, the sequence can be characterized by a simple expression: \( e^{2n} \), where \( n \) classifies the term number beginning at 1.
After establishing a potential pattern, compute the ratios as shown above to confirm geometric progression. Finding the same common ratio across successive terms signals a geometric sequence.
Sequence identification is crucial for categorizing sequences in mathematics and real-world applications, enabling better prediction, analysis, and decision-making.