Problem 39 - Algebra 2 and Trigonometry

Question

In \(31-39,\) write the first five terms of each sequence. $$ a_{5}=\frac{1}{2}, a_{n}=\frac{1}{a_{n-1}} $$

Step-by-Step Solution

Answer

The first five terms are: \( \frac{1}{2}, 2, \frac{1}{2}, 2, \frac{1}{2} \).

Step 1: Understand the Initial Term

The first term of the sequence is given as \( a_5 = \frac{1}{2} \). This means we need to find the value of the first term \( a_1 \) using the provided recursive formula.

Step 2: Apply the Recursive Formula in Reverse

The recursive formula given is \( a_n = \frac{1}{a_{n-1}} \). We will use it in reverse to find previous terms from \( a_5 \). Calculate \( a_4 \) as: \( a_4 = \frac{1}{a_5} = \frac{1}{\frac{1}{2}} = 2 \).

Step 3: Continue Applying the Formula

Repeat the reverse application of the formula to find previous terms. Calculate \( a_3 \) as: \( a_3 = \frac{1}{a_4} = \frac{1}{2} \).

Step 4: Calculate the Next Terms

Find \( a_2 \) using the formula: \( a_2 = \frac{1}{a_3} = \frac{1}{\frac{1}{2}} = 2 \).

Step 5: Find the First Term

Finally, calculate \( a_1 \) as: \( a_1 = \frac{1}{a_2} = \frac{1}{2} \). This traces back to the first term of the sequence.

Step 6: List the First Five Terms

Now that we've calculated from \( a_5 \) back to \( a_1 \), the first five terms of the sequence are: \( a_1 = \frac{1}{2}, a_2 = 2, a_3 = \frac{1}{2}, a_4 = 2, a_5 = \frac{1}{2} \).

Key Concepts

Understanding Sequence Terms

In the world of sequences, **terms** represent individual elements listed in a specific order. Imagine a long chain where each link uniquely connects to the next, forming a beautiful pattern.
The sequence terms are typically denoted using variables and numbers such as \( a_1, a_2, a_3, \cdots, a_n \) to represent their position in the sequence. This way, each term in the sequence is labeled by its position, making it easy to refer back to any specific element. With our example, \( a_5 = \frac{1}{2} \) is given, indicating that the fifth term in this sequence has a value of \( \frac{1}{2} \). This sequential labeling provides a framework to understand how each term connects with the next or previous ones, especially in a pattern as deliberate as a recursive sequence.

Grasping Recursive Formulas

A **recursive formula** is a tool used to define each term of a sequence using the preceding term(s). It's like following a treasure map where each clue leads directly to the next point. Each step depends on the one before it, creating a chain of logical progressions.
In mathematical terms, if you know one term, a recursive formula allows you to determine the next one. Our example uses the formula \( a_n = \frac{1}{a_{n-1}} \). Here, each term is the reciprocal of the term before it. This relationship is key to calculating other terms. Using recursive formulas:

• Generate new terms: Start with a known term and apply the formula to find the next.
• Work backwards or forwards easily if needed.

Recursive sequences like this one rely on concise and continual rules to create a repeated pattern throughout the sequence.

Performing Sequence Calculations

Calculating terms in a recursive sequence often requires sequential steps. Think of it as solving puzzles where each piece seamlessly locks into the next.
To calculate our sequence, we applied the formula in reverse, starting from the known term \( a_5 \) and working back to \( a_1 \). This reverse journey through our sequence was guided by applying the recursive relationship \( a_n = \frac{1}{a_{n-1}} \).Here's how the calculations flow:

• Starting with \( a_5 = \frac{1}{2} \).
• Finding \( a_4 = \frac{1}{a_5} = 2 \).
• Calculating \( a_3 = \frac{1}{a_4} = \frac{1}{2} \).
• Solving \( a_2 = \frac{1}{a_3} = 2 \).
• Determining \( a_1 = \frac{1}{a_2} = \frac{1}{2} \).

This sequence calculation process reveals the first five terms: \( a_1 = \frac{1}{2}, a_2 = 2, a_3 = \frac{1}{2}, a_4 = 2, a_5 = \frac{1}{2} \). Each computation is like a story that unfolds one line at a time, related by the recursive rule.