When we encounter sequences in mathematics, they often follow specific rules or formulas. One such intriguing sequence is defined by the recursive formula f(n + 1) = f(n) – 2. The sequence starts with f(1) = 18, and we are tasked with finding the value of f(5). In this article, we will delve into the world of recursive sequences, understand how they work, and determine the value of f(5) step-by-step. So, buckle up and embark on this mathematical journey with us!
A Sequence Defined by the Recursive Formula f(n + 1) = f(n) – 2
The sequence defined by the recursive formula f(n + 1) = f(n) – 2 is a fascinating mathematical construct. Let’s break down the formula:
f(n) represents the nth term in the sequence.
f(n + 1) denotes the next term in the sequence.
By subtracting 2 from f(n), we generate the next term.
The recursive nature of this formula means that each term in the sequence relies on the previous one. As we progress through the sequence, each value depends on the value that came before it. Now, let’s proceed to find the value of f(5) when f(1) = 18.
Calculating f(5) Step-by-Step
Step 1: Finding f(2)
To calculate f(2), we need to apply the recursive formula f(n + 1) = f(n) – 2 using f(1) = 18 as the initial value:
f(2) = f(1) – 2
f(2) = 18 – 2
f(2) = 16
Step 2: Finding f(3)
Now, we use the result from Step 1 (f(2) = 16) as the new value for f(2) to calculate f(3):
f(3) = f(2) – 2
f(3) = 16 – 2
f(3) = 14
Step 3: Finding f(4)
Continuing the process, we use the value from Step 2 (f(3) = 14) as the new value for f(3) to calculate f(4):
f(4) = f(3) – 2
f(4) = 14 – 2
f(4) = 12
Step 4: Finding f(5)
Finally, we employ the result from Step 3 (f(4) = 12) as the new value for f(4) to calculate f(5):
f(5) = f(4) – 2
f(5) = 12 – 2
f(5) = 10
The Value of f(5)
After following the steps, we have determined that f(5) is equal to 10. Thus, the fifth term in the sequence defined by the recursive formula f(n + 1) = f(n) – 2, with f(1) = 18, is 10.
Exploring the Applications of Recursive Sequences
Recursive sequences like the one we just explored have various applications in mathematics, computer science, and real-life scenarios. They are commonly used in:
Fractals: Recursive sequences play a crucial role in generating mesmerizing fractals, intricate patterns that repeat infinitely.
Algorithms: Many algorithms are based on recursive principles, making them vital in computer science and programming.
Financial Modeling: Recursive sequences are used to model financial systems and forecast trends in economics.
Population Studies: Recursive sequences can be applied to study population growth and predict future demographics.
FAQs
What is the significance of a recursive sequence?
A recursive sequence relies on its previous terms to generate subsequent ones, making it a powerful tool in modeling various phenomena.
Can recursive sequences have different recursive formulas?
Yes, different recursive formulas can define various sequences, each with its distinct pattern.
How do recursive sequences differ from explicit sequences?
Recursive sequences define terms in relation to previous terms, while explicit sequences directly provide the formula to calculate any term.
Are recursive sequences always predictable?
While some recursive sequences exhibit predictable patterns, others may behave chaotically and be harder to predict.
How are recursive sequences represented graphically?
Recursive sequences are often depicted as plots on a graph, with the x-axis representing the term number and the y-axis representing the value.
Can recursive sequences have negative terms?
Yes, depending on the recursive formula, recursive sequences can have negative, positive, or even fractional terms.
Conclusion
In conclusion, the sequence defined by the recursive formula f(n + 1) = f(n) – 2, with f(1) = 18, has been a fascinating journey through the world of mathematics. We explored the recursive nature of the sequence, calculated f(5) step-by-step, and learned about its applications in various fields. Understanding recursive sequences allows us to unravel the complexities of patterns and relationships within our world.
So, the next time you encounter a sequence defined by a recursive formula, remember the steps we took to find f(5) and appreciate the beauty of mathematical patterns.