Problem 30 - Precalculus: Mathematics for Calculus

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \dots\)

Step-by-Step Solution

Answer

The formula for the nth term is \(a_n = \frac{(-1)^n}{3^n}\).

Step 1: Analyze the Pattern

First, observe the signs and numerators stay consistent as 1. The denominators are powers of 3: \(3, 9, 27, 81\) which can be written as \(3^1, 3^2, 3^3, 3^4\). This suggests an exponential formula involving powers of 3.

Step 2: Determine the Sequence Formula

Notice the sign alternates. The sequence suggests each term is \((-1)^n\) times \(1/(3^n)\). Therefore, the general formula for the \(n\)th term is: \[ a_n = \frac{(-1)^n}{3^n} \]

Step 3: Verify the Formula

Apply the formula to initial terms: For \(n = 1\), \(a_1 = \frac{(-1)^1}{3^1} = -\frac{1}{3}\); for \(n = 2\), \(a_2 = \frac{(-1)^2}{3^2} = \frac{1}{9}\). Continued calculations show this formula fits all given terms.

Key Concepts

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a specific pattern or rule. Each number in the list is called a "term" of the sequence. In the given exercise, we start with the terms \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \ldots\). Here, each term is developed from a shared rule that dictates the progression of the sequence. Understanding this rule is essential as it allows us to extend the sequence indefinitely.
When analyzing sequences, look for patterns such as consistent arithmetic (difference between terms) or geometric (ratio between terms) properties. In this example, the pattern emerges as a geometric sequence where the denominator is a power of 3 and the sign alternates. Recognizing these key features is a critical skill in sequence analysis.
The general term or formula for a sequence enables us to find any term without listing them all. This aids in predicting further terms and is particularly useful in applications of mathematical modeling where sequences are prevalent.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. This type of function is prevalent in sequences that increase or decrease exponentially. In our sequence, each term's denominator is an exponential value of 3, expressed as \(3^n\).
It's important to realize how quickly exponential sequences grow or decay, which highlights their significance in various mathematical and real-world contexts. In this exercise, the terms decrease much faster due to the increasing exponent of 3 in the denominator, emphasizing the control the base and exponent hold over the sequence's behavior.

• The base here is 3, which means each step in the sequence involves multiplying the previous term's denominator by another 3.
• When combined with an alternating sign factor, exponential functions can create more complex sequences, showcasing diverse growth patterns.

Recognizing exponential patterns in sequences is a valuable skill, especially as they appear frequently in science and finance.

Alternating Series

Alternating series are distinctive because the sign of the terms changes with each step in the sequence. This shift creates a zigzag pattern, adding complexity to the sequence's behavior. In the sequence from the exercise, the terms alternate from negative to positive.
This alternation is captured by the factor \((-1)^n\), where \(n\) represents the term's position. The expression \((-1)^n\) ensures that all odd terms are negative and even terms are positive. By understanding this mechanism, we can predict and confirm the sign of any subsequent term.
Alternating series often appear in mathematical contexts where converging sums are significant, such as infinite series or oscillating systems. By mastering the alternating sign pattern, students can grasp the nuance and predict outcomes in more advanced mathematical scenarios.