Problem 24 - College Algebra

Question

Write an explicit formula for each sequence. $$0, \frac{1-e^{1}}{1+e^{2}}, \frac{1-e^{2}}{1+e^{3}}, \frac{1-e^{3}}{1+e^{4}}, \frac{1-e^{4}}{1+e^{5}}, \ldots$$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = \frac{1 - e^n}{1 + e^{n+1}} \) for \( n \geq 0 \).

Step 1: Identify the Sequence Pattern

First, observe the terms in the sequence to determine how they are generated. Each term seems to be of the form \( \frac{1 - e^{n}}{1 + e^{n+1}} \), where \( n \) is a positive integer starting from 0 for the first term. We will test this pattern with the given terms.

Step 2: Validate the Pattern

Check if the identified pattern holds true for the provided terms. The first term is 0, which corresponds to \( n = 0 \), so the term becomes \( \frac{1 - e^0}{1 + e^1} = \frac{1 - 1}{1 + e^1} = 0 \). This confirms the pattern for \( n = 0 \). Continue validation with subsequent terms: for \( n = 1 \), the term is \( \frac{1 - e^1}{1 + e^2} \), which matches the second element in the sequence. The same pattern validates all available terms.

Step 3: Write the Explicit Formula

From our validation, we confirm that the sequence can be described by the formula \( a_n = \frac{1 - e^n}{1 + e^{n+1}} \), where \( n \geq 0 \). This explicit formula correctly represents each term of the sequence.

Key Concepts

Understanding the Sequence Pattern

To explore the concept of a sequence pattern, you first need to observe and analyze the terms provided in a sequence. In our case, the sequence is as follows:

• 0
• \( \frac{1-e^{1}}{1+e^{2}} \)
• \( \frac{1-e^{2}}{1+e^{3}} \)
• \( \frac{1-e^{3}}{1+e^{4}} \)
• \( \frac{1-e^{4}}{1+e^{5}} \)

The key to identifying the pattern is recognizing that each term is built from a specific mathematical relationship involving exponents. Notice how the power of \( e \) in the numerator begins at 0 and increases by one in each subsequent term, so does the power of \( e \) in the denominator, though it starts one step ahead.
In essence, recognizing this regular pattern lets us anticipate the construction of future sequence elements, which describes each term in terms of its order within the sequence. Understanding patterns is crucial, especially when you need to develop a formula to describe the sequence explicitly.

Steps of Sequence Validation

Once a pattern appears, it’s important to validate it by checking if it consistently applies across all terms. Validation ensures that the identified sequence pattern is not a mere coincidence but a rule governing term formation. Let’s validate the initial sequence's identified pattern step by step.
Start by substituting different term positions (given by \( n \)) into the suggested pattern \( \frac{1 - e^n}{1 + e^{n+1}} \) and check against the sequence terms. Starting with \( n = 0 \), the expression becomes \( \frac{1 - e^0}{1 + e^1} \), simplifying to 0, which matches the first term in the sequence. For \( n = 1 \), it becomes \( \frac{1 - e^1}{1 + e^2} \), which is exactly the second term. By executing this calculation at each step and confirming its result aligns with the sequence, the sequence pattern is validated.

• Ensures consistency
• Verifies theoretical predictions with actual data
• Enhances comprehension of sequence behavior

Validation is a pivotal process in sequence analysis and is fundamental to writing the explicit formula.

Exploring Algebraic Sequences

An algebraic sequence is expressed as a formula that describes the relationship of any term in the sequence based on its order (usually represented by \( n \)). This explicit formula helps in conveniently identifying any term's value in the sequence without listing all previous terms.
In our scenario, the explicit formula derived from observed patterns and validation is \( a_n = \frac{1 - e^n}{1 + e^{n+1}} \), starting with \( n \geq 0 \). This formula is algebraic due to the use of algebraic functions and operations (like exponentiation and division).

• Makes calculation of any term direct
• Relies on understanding of exponents and algebraic operations
• Offers insight into sequence trends

Understanding how to transform observed sequence patterns into explicit algebraic expressions is key in mathematics, enabling efficient analysis and manipulation of sequences.