Problem 1 - Algebra 2 and Trigonometry

Question

Nichelle said that sequence of numbers in which each term equals half of the previous term is a finite sequence. Randi said that is an infinite sequence. Who is correct? Justify your answer.

Step-by-Step Solution

Answer

Randi is correct; the sequence is infinite, as it can continue indefinitely.

Step 1: Understanding the Problem

The problem asks us to analyze a sequence where each term is half of the previous term. We are to determine if this sequence is finite or infinite and justify our determination.

Step 2: Defining the Sequence

Let's consider a sequence where the first term is denoted as \(a_1\). Each subsequent term \(a_{n+1}\) is defined by the rule \(a_{n+1} = \frac{1}{2} a_n\). This means every term is half of the previous one.

Step 3: Analyzing Finite vs Infinite

A finite sequence has a limited number of terms, while an infinite sequence continues indefinitely. For this problem, since each term is generated by halving the previous term, no matter how small the terms get, the sequence continues forever without reaching a stopping point.

Step 4: Conclusion on the Nature of the Sequence

Given that the generation rule \(a_{n+1} = \frac{1}{2} a_n\) allows us to continue computing the next term indefinitely, this sequence does not have a natural endpoint. Thus, it is an infinite sequence.

Key Concepts

Understanding Geometric Sequences

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the case of the sequence mentioned in the exercise, the common ratio is \( \frac{1}{2} \). This means each term is half the value of the one before it.
For example, if the first term \(a_1\) is given as 8, the sequence would continue as 8, 4, 2, 1, \(\frac{1}{2}\), and so forth, decreasing with each subsequent term. The defining characteristic of geometric sequences is that to go from one term to the next, you consistently use the same multiplication factor, which allows for predictable behavior.
Geometric sequences can be finite or infinite, depending on how they are defined. In this context, because the sequence continues indefinitely without reaching a termination point, the sequence is infinite. If it had been limited by a specific number of terms, it would be considered finite.

Defining a Sequence

A sequence is simply an ordered list of numbers. Each number in the list is referred to as a term. Sequences can be like an instruction manual on how to create each term based on the previous ones or some initial value.
The notation for sequences often involves denoting terms with symbols like \(a_n\), where \(n\) indicates the term number in the order. In our example, the first term \(a_1\) sets the starting point. The rule \(a_{n+1} = \frac{1}{2} a_n\) tells us how to find the subsequent terms.

• When defining a sequence, you typically need a starting point, like \(a_1 = 1\).
• Then apply the rule in a consistent manner to find each subsequent term.

Understanding how sequences are defined helps in predicting future terms or analyzing their behavior, such as determining whether they are finite or infinite.

Analyzing Sequences

Sequence analysis involves exploring the characteristics and behaviors of sequences to draw conclusions about their nature. One of the first aspects to consider is whether a sequence is finite or infinite.
A finite sequence has a definite number of terms. Think of a countdown from ten to zero. Once you reach zero, the sequence stops. An infinite sequence, conversely, doesn't really stop; instead, it continues endlessly, either growing larger or shrinking towards zero, as in our example where each term is half of its predecessor.
When a sequence, such as the one in the exercise, continues without a definitive endpoint because halving can continue indefinitely, it's recognized as infinite. This form of analysis allows us to understand how sequences behave, what patterns they form, and possible applications for such mathematical structures in real-world scenarios.