Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identifying the Pattern

The sequence given is \( e^2, e^4, e^6, e^8, \dots \). To determine if it is geometric, we need to check if there is a consistent pattern of multiplication (common ratio) between the terms.

Step 2: Calculating the Ratio

In a geometric sequence, the ratio between consecutive terms is constant. Calculate the ratio by dividing the second term by the first term: \( \frac{e^4}{e^2} = e^{4-2} = e^2 \).

Step 3: Checking Consistency of Ratio

Verify if the same ratio applies to the other consecutive terms. Compute the ratio \( \frac{e^6}{e^4} \): \( \frac{e^6}{e^4} = e^{6-4} = e^2 \); and \( \frac{e^8}{e^6} = e^{8-6} = e^2 \). The ratio is consistent.

Step 4: Conclusion on Sequence Type

Since the ratio \( e^2 \) is constant across all pairs of consecutive terms, the sequence is indeed geometric.

Key Concepts

Common Ratio

A geometric sequence is defined by its common ratio. The common ratio is the constant factor by which each term in the sequence is multiplied to get the next term. For example, in a sequence like 2, 4, 8, 16, each term is obtained by multiplying the previous term by 2. Here, the common ratio is 2.
In our given sequence, \( e^2, e^4, e^6, e^8 \), we determine the common ratio by examining the ratio between consecutive terms. To find the common ratio, divide one term by the previous one. So, for the terms \( e^4 \) and \( e^2 \), their ratio is \( \frac{e^4}{e^2} = e^{4-2} = e^2 \).

• The common ratio in a geometric sequence helps identify the sequence type.
• It remains the same for all pairs of consecutive terms throughout the sequence.
• In our problem, the common ratio is \( e^2 \), which confirms the sequence's geometric nature.

Exponential Function

Exponential functions are mathematical expressions where the variable is in the exponent. They are often used in sequences that grow or decay at constant rates. For instance, in an exponential function like \( a^n \), \( a \) is a constant, and \( n \) is the variable exponent.
In the sequence \( e^2, e^4, e^6, e^8 \), each term can be thought of as an exponential function where the exponent's rate of increase is consistently 2. This means the sequence's growth is directly connected to the properties of exponential functions.

• Exponential sequences appear often in phenomena of rapid growth or decay.
• They are closely related to geometric sequences, especially when exponential constants are bases of the terms.
• In our exercise, \( e^n \) represents such an exponential function where \( n \) is increased in steps of 2 in each consecutive term.

Sequence Pattern

Understanding the pattern in sequences is key to identifying their type. In a geometric sequence, the pattern is identified by a consistent multiplicative factor, known as the common ratio. Recognizing this pattern helps in predicting future terms.
The sequence \( e^2, e^4, e^6, e^8 \) follows a clear pattern: each term is derived by multiplying the previous one by \( e^2 \). This indicates a constant exponential increase between terms.

• To spot a sequence pattern, compare consecutive terms regularly.
• In geometric sequences, the pattern reflects a specific, predictable growth or shrinkage path due to the common ratio.
• For our sequence, the pattern is evident through both the exponential increase and the calculation of the common ratio \( e^2 \).