A Sequence is Defined The Recursive Formula f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?

In the realm of mathematics, sequences are fascinating entities that unfold in a patterned manner, often following predefined rules. One such method of defining sequences is through recursive formulas, where each term is determined by preceding terms according to a specific rule. In this article, we delve into the intricacies of a sequence is defined by the recursive formula f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)? With an initial condition of f(1) = 18, we embark on a journey to unveil the value of f(5), navigating through the recursive nature of the sequence.

Understanding Recursive Formulas

What are Recursive Formulas?

Recursive formulas, such as “a sequence is defined by the recursive formula f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?”, are vital mathematical tools. They offer a systematic approach to deriving sequences, crucial not only in mathematics but also in computer science, where they facilitate algorithmic development and problem-solving strategies.

Of course! We can find the value of f(5) using the given recursive formula and f(1) = 18.

Here’s how:

1. Start with the given information: f(1) = 18.

2. Use the recursive formula to express f(2) in terms of f(1): f(2) = f(1) – 2 = 18 – 2 = 16.

3. Now, express f(3) in terms of f(2): f(3) = f(2) – 2 = 16 – 2 = 14.

4. Similarly, express f(4) in terms of f(3): f(4) = f(3) – 2 = 14 – 2 = 12.

5. Finally, express f(5) in terms of f(4): f(5) = f(4) – 2 = 12 – 2 = 10.

I’d be happy to provide more information! What specifically would you like to know about the sequence or the process of finding f(5)? Here are some options:

General information about the sequence:

• Explicit formula: While the recursive formula is given, there is also an explicit formula for this sequence. Would you like to see how to derive it and what it looks like?
• Properties: This sequence has interesting properties like being an arithmetic sequence with a common difference of -2. Do you want to explore this further?
• Applications: Are you curious about potential real-world applications of this type of sequence?

More details about finding f(5):

• Alternative methods: Would you like to see alternative ways to find f(5), like using iteration or a table?
• Generalization: Can you see how this method can be used to find any term in the sequence given the first term and the recursive formula?
• Connection to other concepts: Are you interested in how this sequence relates to other mathematical concepts like linear functions or difference equations?

Deconstructing f(n + 1) = f(n) – 2

The recursive formula “a sequence is defined by the recursive formula f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?” signifies that each term in the sequence is obtained by subtracting 2 from the preceding term. This simple yet elegant rule governs the progression of the sequence, shaping its evolution with each iteration. As the sequence advances, each subsequent term reflects a decrement of 2 from its predecessor, leading to a consistent pattern of decrease throughout the sequence.

Solving the Sequence

Establishing the Initial Condition

Before delving into the sequence, it is crucial to establish the initial condition provided: f(1) = 18. This initial term serves as the starting point for our exploration of the sequence. Understanding the starting value enables us to trace the sequence’s development accurately and predict its subsequent terms with confidence.

Calculating f(2)

To find the value of f(2), we apply the recursive formula:

f(2) = f(1) – 2

Substituting f(1) with the given value of 18:

f(2) = 18 – 2 f(2) = 16

Determining f(3) and f(4)

Continuing the pattern, we can calculate the values of f(3) and f(4) using the recursive formula:

f(3) = f(2) – 2 f(4) = f(3) – 2

Substituting the previously calculated values:

f(3) = 16 – 2 f(3) = 14

f(4) = 14 – 2 f(4) = 12

Finding f(5)

With the values of f(1), f(2), f(3), and f(4) determined, we can proceed to find f(5) using the recursive formula:

f(5) = f(4) – 2

Substituting the value of f(4):

f(5) = 12 – 2 f(5) = 10

Conclusion