Fibonacci Sequence F(9) Calculation And Problem Solution

by Mei Lin 57 views

The Fibonacci sequence, a fascinating mathematical concept, is defined by the recurrence relation F(n) = F(n-1) + F(n-2). Guys, this means that each number in the sequence is the sum of the two preceding numbers. We're given that F(7) = 13 and F(8) = 21, and our mission, should we choose to accept it, is to figure out which statement about the sequence is true. So, let's dive in and unravel this numerical mystery!

Decoding the Fibonacci Sequence

To really understand this Fibonacci sequence problem, let's break down the basics. The sequence starts with 0 and 1, and from there, we add the previous two numbers to get the next one. So, it goes like this: 0, 1, 1, 2, 3, 5, 8, and so on. The formula F(n) = F(n-1) + F(n-2) is just a fancy way of saying this. It tells us that to find any term in the sequence (F(n)), we simply add the two terms before it (F(n-1) and F(n-2)). We're given two consecutive Fibonacci numbers, F(7) = 13 and F(8) = 21. This is a crucial piece of information because it allows us to calculate other terms in the sequence. Think of it as having two puzzle pieces that fit together perfectly, giving us the ability to build more of the Fibonacci picture. The beauty of the Fibonacci sequence lies in its simplicity and its surprising appearance in various natural phenomena, from the arrangement of leaves on a stem to the spirals of a sunflower. Understanding the core concept of adding the previous two terms unlocks a world of mathematical exploration. So, with this foundational knowledge, we can confidently tackle the challenge of finding F(9) and verifying the given statements. Let's keep this energy going, guys, and crack this Fibonacci code!

Finding F(9): The Next Step in the Sequence

Now that we've got a handle on the Fibonacci sequence definition, let's zero in on finding F(9). We know that F(8) = 21 and F(7) = 13. Using the formula F(n) = F(n-1) + F(n-2), we can find F(9) by adding F(8) and F(7). So, F(9) = F(8) + F(7) = 21 + 13 = 34. Boom! We've found F(9). This calculation is super straightforward, but it's a critical step in solving the problem. It highlights the power of the recursive definition of the Fibonacci sequence – each term builds upon the previous ones. This simple addition unlocks a new value in the sequence, allowing us to explore further. Now, with F(9) = 34 in our arsenal, we can evaluate the given options and see which one rings true. It's like we've found the key to the next level in the Fibonacci game, and we're ready to move forward with confidence. Let's keep this momentum going, guys, and nail this problem!

Evaluating the Options: Which Statement Holds True?

Alright, guys, we've cracked the code for F(9), and now it's time to put on our detective hats and evaluate the given options. We need to see which statement aligns with our calculations and understanding of the Fibonacci sequence. Let's break down each option methodically:

  • A. F(9) = 34: This is exactly what we calculated! So, this option is looking pretty good right off the bat. It directly matches our finding, making it a strong contender for the correct answer. But let's not jump to conclusions just yet; we need to analyze the other options to be absolutely sure.
  • B. F(9) = 32: This doesn't match our calculation of F(9) = 34, so we can confidently eliminate this option. It's a clear mismatch, and we can move on with our investigation.
  • C. F(15) = 34: To check this, we'd need to calculate F(15), which would involve several more steps in the Fibonacci sequence. However, since the Fibonacci sequence increases, and we know F(9) = 34, it's highly unlikely that F(15) would also be 34. The sequence grows exponentially, so later terms will be significantly larger. This allows us to logically deduce that this option is incorrect without doing a full calculation, saving us precious time and effort.
  • D. F(6) = 13: We know F(7) = 13, so F(6) would have to be a smaller number. Remember, the Fibonacci sequence is built by adding the previous two terms. So, 13 would be the result of adding F(5) and F(6), meaning F(6) must be less than 13. Therefore, this option is also incorrect.

By systematically evaluating each option, we've confirmed that A. F(9) = 34 is the only true statement. We've not only found the value of F(9) but also used our understanding of the sequence to eliminate the other possibilities. This is a powerful approach to problem-solving, and we've nailed it!

Confirming the Solution: Option A is the Winner!

After carefully analyzing all the options, the clear winner is A. F(9) = 34. We calculated F(9) directly using the Fibonacci sequence formula, and it perfectly matches this statement. The other options were demonstrably false, either through direct comparison or logical deduction based on the properties of the Fibonacci sequence. This methodical approach ensures we've arrived at the correct answer with confidence. We didn't just guess; we understood the underlying principle and applied it strategically. This is the essence of mathematical problem-solving, guys. We've not only found the answer but also solidified our understanding of the Fibonacci sequence. So, give yourselves a pat on the back – you've conquered this Fibonacci challenge!

In conclusion, by understanding the Fibonacci sequence and applying the recurrence relation F(n) = F(n-1) + F(n-2), we successfully determined that F(9) = 34, making option A the correct answer. This problem highlights the beauty and elegance of mathematical sequences and the power of logical deduction. Keep exploring, keep questioning, and keep unlocking the mysteries of mathematics, guys!