Problem 16 - Algebra and Trigonometry

Question

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=\frac{1}{1+a_{n-1}} \quad \text { and } \quad a_{1}=1 $$

Step-by-Step Solution

Answer

The first five terms are: 1, \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{3}{5}\), \(\frac{5}{8}\).

Step 1: Understand the Recursive Formula and Initial Condition

The sequence is defined recursively with the formula \( a_n = \frac{1}{1 + a_{n-1}} \). This means that each term is calculated based on the previous term. We are also given the starting value (or the first term) of the sequence, \( a_1 = 1 \).

Step 2: Calculate the Second Term

To find \( a_2 \), use the formula with \( a_1 \): \[ a_2 = \frac{1}{1 + a_1} = \frac{1}{1 + 1} = \frac{1}{2}. \] So, \( a_2 = \frac{1}{2} \).

Step 3: Calculate the Third Term

Use \( a_2 \) to find \( a_3 \): \[ a_3 = \frac{1}{1 + a_2} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}. \] Thus, \( a_3 = \frac{2}{3} \).

Step 4: Calculate the Fourth Term

Use \( a_3 \) to find \( a_4 \): \[ a_4 = \frac{1}{1 + a_3} = \frac{1}{1 + \frac{2}{3}} = \frac{1}{\frac{5}{3}} = \frac{3}{5}. \] Therefore, \( a_4 = \frac{3}{5} \).

Step 5: Calculate the Fifth Term

Use \( a_4 \) to find \( a_5 \): \[ a_5 = \frac{1}{1 + a_4} = \frac{1}{1 + \frac{3}{5}} = \frac{1}{\frac{8}{5}} = \frac{5}{8}. \] Hence, \( a_5 = \frac{5}{8} \).

Key Concepts

Sequence Operations

Sequence operations are essential when dealing with recursively defined sequences. Essentially, each new term in the sequence depends on the calculation involving the previous one. This step-by-step dependency establishes a chain where the understanding and correctness of each part affect the entire sequence.

In the given exercise, our main operation is the recursive formula applied repeatedly to determine each next term. The sequence operation uses the expression \( a_n = \frac{1}{1 + a_{n-1}} \), which defines how each term is generated from the one preceding it. This operation involves division and addition, combined in a specific order:

• Add 1 to the previous term.
• Take the reciprocal of the result.

These specific steps ensure each term gradually converges towards a stable value as more terms are calculated. Exploring these operations helps illuminate the unique behavior and properties of recursively defined sequences.

Initial Conditions

Initial conditions play a crucial role in determining the behavior of a recursive sequence. They serve as the starting point from which all subsequent terms are generated. In other words, they "kick off" the process.

In our example problem, the initial condition is \( a_1 = 1 \). This first term essentially gets the sequence rolling, allowing the recursive formula to build subsequent terms. Initial conditions must be known to apply the sequence operations as they provide the first value needed to begin calculations.

• By using the initial value, we calculate the next term.
• Each subsequent term uses its predecessor to determine its value effectively.

Without the initial condition, it would be impossible to start creating sequence terms. The initial condition's choice influences the entire sequence and can lead to vastly different outcomes under different scenarios.

Recursive Formulas

Recursive formulas define the way each term in a sequence is related to one or more of its predecessors. In other words, they provide a rule for moving from one term to the next within the sequence. This makes them a foundational aspect of many mathematical and computational sequences.

For the exercise, the sequence is structured by the recursive formula \( a_n = \frac{1}{1 + a_{n-1}} \). In simpler terms, to find \( a_n \), you use the previous term \( a_{n-1} \): add 1 to it, then take the reciprocal. This formula sets up a pattern that continues indefinitely, producing a chain of values that depends entirely on the succeeding from previous values.

• A typical recursive sequence must include a base case, the initial condition we already talked about.
• Recursive definitions require a function or rule to determine each term based on the previous ones.

Understanding recursive formulas is crucial to navigate through sequences effectively, as they dictate the overall design of your output.

Sequence Terms

Sequence terms are the actual values generated at each step in a recursive sequence, representing the output resulting from applying recursive formulas and operations. They are the backbone of understanding sequences.

In our example, regular application of the recursive formula produces a set of numbers: These are

• \( a_1 = 1 \)
• \( a_2 = \frac{1}{2} \)
• \( a_3 = \frac{2}{3} \)
• \( a_4 = \frac{3}{5} \)
• \( a_5 = \frac{5}{8} \)

Each term is dependent on its predecessor, and understanding each term's relation helps visualize how sequences evolve. Sequence terms can provide insight into patterns, convergence behaviors, or even lead to identifying general formulas or expressions for the sequence. These tangible results of sequence operations and recursive formulas are ultimately what solve or describe the problem.