Discover the Magnetic Power of Recursive Sequences: F(5) in the Mind-Blowing World of F(N + 1) = F(N)

A Sequence Is Defined By The Recursive Formula F (N + 1) = F(N) – 2. If F(1) = 18, What Is F(5)?

A sequence defined by the recursive formula F(N + 1) = F(N) - 2. Given F(1) = 18, find F(5).

A sequence is a fascinating mathematical concept that involves a set of numbers arranged in a specific order. It often sparks curiosity and leaves us wondering about the patterns and relationships hidden within. One way to define a sequence is through a recursive formula, where each term is determined by the previous one. In this particular case, we are given the recursive formula F (N + 1) = F(N) – 2. Intriguingly, the initial value of F(1) is provided as 18. Now, let's embark on a journey to discover the value of F(5) and unravel the secrets this sequence holds.

Introduction

Sequences play a crucial role in mathematics, and they are often defined by recursive formulas that allow us to find the value of each term based on the previous one. In this article, we will explore a sequence defined by the recursive formula F(N + 1) = F(N) – 2, with an initial condition given as F(1) = 18. By applying the recursive formula multiple times, we can determine the value of F(5).

Understanding the Recursive Formula

The given recursive formula, F(N + 1) = F(N) – 2, tells us that to find any term in the sequence, we need to subtract 2 from its preceding term. This formula allows us to create a chain of numbers where each term is obtained by subtracting 2 from the previous one.

Finding F(2)

To find F(2), we can substitute the value of N as 1 into the recursive formula. Thus, F(2) = F(1) – 2. Given that F(1) = 18, we can calculate F(2) as follows: F(2) = 18 – 2 = 16.

Finding F(3)

Similar to finding F(2), to find F(3), we substitute N as 2 in the formula: F(3) = F(2) – 2. We have previously determined that F(2) = 16, so F(3) can be calculated as follows: F(3) = 16 – 2 = 14.

Finding F(4)

Following the same approach as before, we substitute N as 3 in the formula: F(4) = F(3) – 2. Using our previous result of F(3) = 14, we can calculate F(4) as follows: F(4) = 14 – 2 = 12.

Finding F(5)

Finally, to find F(5), we substitute N as 4 in the formula: F(5) = F(4) – 2. By using our previous result of F(4) = 12, we can calculate F(5) as follows: F(5) = 12 – 2 = 10.

Conclusion

By applying the recursive formula F(N + 1) = F(N) – 2, with an initial condition of F(1) = 18, we have successfully determined the value of F(5) to be 10. This demonstrates how recursive formulas can be used to find the values of terms in a sequence based on a given pattern. Understanding sequences and their recursive formulas is essential in various mathematical fields, including algebra, calculus, and number theory.

Introduction: Understanding the concept of a sequence defined by a recursive formula

In mathematics, a sequence is a set of numbers arranged in a specific order. These sequences can be defined using different formulas or patterns. One type of sequence is called a recursive sequence, which is defined by a recursive formula. A recursive formula expresses each term in the sequence in terms of one or more previous terms.

Defining the recursive formula: Explaining the given formula F(N + 1) = F(N) – 2 and its significance

The recursive formula given in this problem is F(N + 1) = F(N) – 2. This means that each term in the sequence is obtained by subtracting 2 from the previous term. The significance of this formula lies in its ability to generate a sequence where each term is dependent on the previous term. It allows us to establish a relationship between the terms and observe a pattern as the sequence progresses.

Initial condition: Stating the value of F(1) as 18, the starting point of the sequence

To find the value of F(5), we need to calculate the terms leading up to it. The initial condition given in the problem is F(1) = 18, which indicates that the first term in the sequence is 18. This acts as the starting point for our calculations.

Calculating F(2): Using the recursive formula to find the value of F(2) in the sequence

Now, let's apply the recursive formula F(N + 1) = F(N) – 2 to find the value of F(2). We substitute N = 1 into the formula, as F(2) is the term that follows F(1).

F(2) = F(1) – 2

F(2) = 18 – 2

F(2) = 16

Calculating F(3): Applying the recursive formula to find the value of F(3) in the sequence

Next, we continue the pattern by using the recursive formula to calculate F(3). We substitute N = 2 into the formula, as F(3) follows F(2).

F(3) = F(2) – 2

F(3) = 16 – 2

F(3) = 14

Calculating F(4): Continuing the pattern by using the recursive formula to find the value of F(4) in the sequence

Continuing with the recursive formula, we can now find the value of F(4). Substituting N = 3 into the formula, as F(4) follows F(3).

F(4) = F(3) – 2

F(4) = 14 – 2

F(4) = 12

Calculating F(5): Extending the recursive formula further to determine the value of F(5) in the sequence

To calculate F(5), we extend the application of the recursive formula. Substituting N = 4 into the formula, as F(5) follows F(4).

F(5) = F(4) – 2

F(5) = 12 – 2

F(5) = 10

Discussion: Analyzing the pattern observed in the calculated values and its relation to the initial condition

From the calculations, we observe a clear pattern in the sequence. Each term is obtained by subtracting 2 from the previous term. This consistent decrement of 2 creates a linear progression in the sequence. Additionally, we can see that the value of each term decreases by 2 with each step.

Furthermore, we can establish a direct relationship between the initial condition F(1) = 18 and the subsequent terms. By subtracting 2 repeatedly, we obtain the values of F(2), F(3), F(4), and finally F(5). This showcases how the recursive formula connects each term to the one that precedes it, allowing us to generate the entire sequence.

Final result: Revealing the value of F(5) based on the given recursive formula and initial condition

Based on the given recursive formula F(N + 1) = F(N) – 2 and the initial condition F(1) = 18, we have calculated the value of F(5) to be 10. This signifies that the fifth term in the sequence generated by the recursive formula is 10.

Conclusion: Summarizing the approach taken to find F(5) and emphasizing the importance of recursive formulas in understanding sequences

In this analysis, we utilized the recursive formula F(N + 1) = F(N) – 2 to find the value of F(5) in the given sequence. By applying the formula successively and considering the initial condition F(1) = 18, we calculated the values of F(2), F(3), F(4), and finally F(5).

This process allowed us to observe a pattern in the sequence, where each term is derived by subtracting 2 from the previous term. We also highlighted the significance of recursive formulas in understanding sequences, as they establish a relationship between terms and enable us to generate the entire sequence based on a given starting point.

In conclusion, the value of F(5) in the sequence defined by the recursive formula F(N + 1) = F(N) – 2 and with the initial condition F(1) = 18 is 10. This analysis demonstrates how recursive formulas can be utilized to unravel the progression of sequences and emphasizes their importance in mathematical understanding.

In this problem, we are given a sequence defined by the recursive formula F(N + 1) = F(N) - 2. We need to determine the value of F(5) when F(1) is equal to 18.

Let's break down the sequence step by step:

Given: F(1) = 18
Using the recursive formula, we can find the next term in the sequence:

F(2) = F(1) - 2 = 18 - 2 = 16

Similarly, we can find the next term:

F(3) = F(2) - 2 = 16 - 2 = 14

Continuing with the pattern:

F(4) = F(3) - 2 = 14 - 2 = 12

And finally we can find F(5):

F(5) = F(4) - 2 = 12 - 2 = 10

Therefore, when F(1) is 18, the value of F(5) is 10.

In conclusion, the sequence defined by the recursive formula F(N + 1) = F(N) - 2, with F(1) = 18, leads us to the value of F(5) being 10.

Thank you for visiting our blog today! In this article, we have delved into the concept of sequences and explored a specific sequence defined by a recursive formula. By understanding this formula and the given initial condition, we were able to determine the value of the fifth term in the sequence.

Before we dive into the solution, let's recap what we have learned so far. A sequence is an ordered list of numbers that follow a specific pattern or rule. In this case, our sequence is defined by the recursive formula F(N + 1) = F(N) - 2. This means that each term in the sequence can be calculated by subtracting 2 from the previous term.

Now, let's apply this formula to find the value of the fifth term, F(5). We are given the initial condition F(1) = 18. To calculate F(5), we need to work our way up from F(1) by repeatedly applying the recursive formula. Starting with F(1) = 18, we subtract 2 to get F(2), then subtract 2 again to get F(3), and so on until we reach F(5).

By following this process, we find that F(5) is equal to 10. Therefore, the fifth term in the sequence defined by the recursive formula F(N + 1) = F(N) - 2, with the initial condition F(1) = 18, is 10. It is fascinating how a simple formula can generate such a pattern and allow us to determine the value of any term in the sequence.

We hope you found this article informative and enjoyed exploring the concept of recursive sequences. If you have any further questions or would like to explore more mathematical topics, feel free to browse through our blog. Thank you once again for visiting, and we hope to see you again soon!

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Discover the Magnetic Power of Recursive Sequences: F(5) in the Mind-Blowing World of F(N + 1) = F(N) – 2