Problem 18 - Algebra and Trigonometry

Question

Determine whether the sequence is geometric. If 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: Identify the Nature of the Sequence

A sequence is geometric if the ratio of every successive term to the previous term is constant. We will calculate this ratio for the given sequence.

Step 2: Calculate the Common Ratio

The sequence given is \( e^2, e^4, e^6, e^8, \dots \). Calculate the common ratio \( r \) by dividing the second term by the first term. \[ r = \frac{e^4}{e^2} = e^{4-2} = e^2 \] Repeat this for the next terms to confirm that the ratio remains constant. \[ r = \frac{e^6}{e^4} = e^{6-4} = e^2 \] \[ r = \frac{e^8}{e^6} = e^{8-6} = e^2 \]

Step 3: Conclude the Nature of the Sequence

Since the ratio \( r \) is constant and equal to \( e^2 \) for all successive terms, this sequence is confirmed to be geometric.

Step 4: State the Common Ratio

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

Key Concepts

Understanding Common Ratio

A common ratio is a key element of a geometric sequence. It is the factor by which we multiply a term to obtain the next term in the sequence. To find a common ratio, simply divide any term by the previous term.
In our example sequence:

• The first term is \(e^2\)
• The second term is \(e^4\)
• So, the common ratio \( r \) is: \( r = \frac{e^4}{e^2} = e^{4-2} = e^2 \)

This shows that each term is \(e^2\) times the previous one. Finding a common ratio helps determine if a sequence is geometric.

Sequence Analysis Techniques

Sequence analysis involves studying the elements and structure of a sequence to determine its type and characteristics.
For geometric sequences, analysis typically focuses on identifying the common ratio to confirm if the sequence maintains the same multiplicative relationship.
In our sequence:

• Checking the ratio between consecutive terms like \( \frac{e^6}{e^4} = e^2 \)
• And \( \frac{e^8}{e^6} = e^2 \)

These repeated calculations show consistency of the common ratio, affirming the sequence's geometric nature.

Exploring Mathematical Sequences

Mathematical sequences can vary in type, such as arithmetic or geometric. Each type has unique properties:

• Arithmetic sequences have a constant difference between terms.
• Geometric sequences have a constant common ratio.

Recognizing these differences helps in assessing what sequence type you're dealing with.
Our example given is a geometric sequence, highlighted by its consistent ratio \(e^2\). Understanding these foundational concepts of sequences enhances problem-solving skills in mathematics and assists in identifying patterns and predicting future terms effectively.