Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, 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 \(a_n = \frac{1-e^n}{1+e^{n+1}}\).

Step 1: Identify the Terms of the Sequence

Observe the given sequence: \(0, \frac{1-e}{1+e^2}, \frac{1-e^2}{1+e^3}, \frac{1-e^3}{1+e^4}, \frac{1-e^4}{1+e^5}, \ldots\). Each term can be expressed in the form \(f(n) = \frac{1-e^{n}}{1+e^{n+1}}\). This suggests that for each term, the pattern involves using increasing powers of \(e\).

Step 2: Establish a General Formula

From the pattern observed, notice that for \(n=1\), we have \(a_1 = \frac{1-e^1}{1+e^2}\), and for \(n=2\), we have \(a_2 = \frac{1-e^2}{1+e^3}\). Thus, the general explicit formula for the sequence seems to be given by \(a_n = \frac{1-e^n}{1+e^{n+1}}\).

Step 3: Write the Explicit Formula

Based on the established pattern, the explicit formula for the sequence can be formally written as \(a_n = \frac{1-e^{n}}{1+e^{n+1}}\), where \(n\) starts from 1 and goes to the number of terms you need.

Key Concepts

Sequences in Mathematics

Mathematical sequences are ordered lists of numbers that follow a particular rule or pattern. These sequences can be finite, with a specific number of terms, or infinite, continuing indefinitely. Each member of a sequence is called a term, and the position of a term within the sequence is typically denoted with the variable \(n\). Sequences appear in various fields of mathematics, including algebra, calculus, and number theory, providing a foundational basis for analyzing numerical relationships.
The primary goal when working with sequences is to identify the rule that defines how the terms are generated. This rule can be established through observation or using explicit formulas that describe the position of each term in the sequence using mathematical expressions. Comprehending sequences is crucial for solving problems that involve predicting future terms or summing the terms of a sequence.

Patterns in Sequences

In sequences, patterns signify the repeated characteristics or rules governing the sequence's progression. By analyzing these patterns, mathematicians can predict upcoming terms efficiently. For instance, some common patterns are arithmetic, geometric, and others that involve more complex relationships like the one in our exercise.
The sequence in the original exercise displays a specific kind of pattern. Each term is expressed as a fraction involving exponential terms. Noticing this pattern helps in deducing an explicit formula. Understanding these patterns allows one to convert sequences into a predictable and manageable format, making mathematical calculations more straightforward.

• Recognizing whether a sequence involves arithmetic or geometric characteristics helps in identifying common differences or ratios.
• A sequence's pattern may also involve more sophisticated algebraic or exponential forms, as seen in the given example.

Patterns in sequences aren't always easy to identify, but with practice, recognizing them becomes simpler.

Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators (like addition and multiplication) that represent values or relationships. In the context of sequences, such expressions are used to formulate rules or formulas from the observed patterns in the sequence.
The explicit formula derived in the given solution uses such mathematical expressions to describe each term of the sequence. For the exercise, the expression \(a_n = \frac{1-e^n}{1+e^{n+1}}\) was identified to define the sequence's terms. Each component of this expression, like the bases and powers of \(e\), plays a crucial role in establishing the relationship between consecutive terms.

• Expressions help in simplifying calculations by providing a single formula representing any term in the sequence.
• They serve as essential tools in defining the general term, aiding in predicting and calculating terms without listing the sequence manually.

A strong grasp of mathematical expressions is vital for mastering different types of sequences and series.