Problem 27 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \dfrac{2}{3} \)

Step-by-Step Solution

Answer

The first five terms of the sequence are \( \dfrac{2}{3}, \dfrac{2}{3}, \dfrac{2}{3}, \dfrac{2}{3}, \dfrac{2}{3} \).

Step 1: Identify The Type of Sequence

Examine the sequence formula \( a_n = \dfrac{2}{3} \). It is clear that this represents a constant sequence as \( a_n \) does not depend on the value of \( n \).

Step 2: Write Out the First Five Terms

Because the sequence is constant every term is the same, specifically, every term is \( a_n = \dfrac{2}{3} \). Thus, the first five terms are all equal to \( \dfrac{2}{3} \).

Key Concepts

Constant Sequence

When we talk about a constant sequence, we mean that all terms in that sequence are the same. This occurs when the formula used to define the sequence remains unchanged regardless of the position of the term.
If you look at the sequence formula in the exercise, it is given as \( a_n = \frac{2}{3} \). No matter what number you substitute for \( n \), the result is always \( \frac{2}{3} \). This makes it a constant sequence.
Think of it like a straight road with no bumps or turns, every section looks the same and therefore the terms do not change as you move along.

• A constant sequence is represented by the same number, repeated over and over.
• The formula is independent of \( n \), which indicates its position in the sequence.
• In a constant sequence like in this problem, every term equals \( \frac{2}{3} \).

Understanding constant sequences is essential as they are the simplest type of sequence, showing no variation.

Terms of a Sequence

Understanding the terms of a sequence is crucial for examining the sequence's properties. In sequences, terms are the individual elements that make up the sequence.
In this exercise, we are most interested in the first few terms to see the starting values and any patterns.
For the constant sequence \( a_n = \frac{2}{3} \), the terms are drawn from substituting \( n \) into the formula.
Since it’s constant, let's break the initial terms down:

• The first term (when \( n = 1 \)) is \( \frac{2}{3} \)
• The second term (when \( n = 2 \)) is \( \frac{2}{3} \)
• The third term (when \( n = 3 \)) is \( \frac{2}{3} \)
• The fourth term (when \( n = 4 \)) is \( \frac{2}{3} \)
• The fifth term (when \( n = 5 \)) is \( \frac{2}{3} \)

Each term remains the same, illustrating the concept of a constant sequence. This uniformity in terms helps to predict and describe the sequence with minimal effort.

Sequence Formula

The sequence formula is like a handmade recipe for finding each term in a sequence. It tells you exactly how to compute each term based on its position, \( n \), in the sequence.
In this exercise, the formula \( a_n = \frac{2}{3} \) is given. This simple formula indicates that no matter what value \( n \) is, the term will always be \( \frac{2}{3} \).
Formulas can vary greatly, from simple to complex ones, depending on how the terms in a sequence are connected.

• Sequence formulas can be used to predict future terms without having to manually calculate each one.
• A formula that doesn’t change with \( n \) usually represents a constant sequence.

Understanding how to read and use these formulas is key to analyzing and determining the behavior of sequences. Once you grasp the formula, the sequence becomes a lot easier to understand.