What is the 13th Fibonacci number?
Understand the Problem
The question is asking for the 13th number in the Fibonacci sequence. The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers. We need to calculate the sequence up to the 13th number.
Answer
144
Answer for screen readers
144
Steps to Solve
- Define the Fibonacci Sequence
The Fibonacci sequence is defined as follows: $F(0) = 0$ $F(1) = 1$ $F(n) = F(n-1) + F(n-2)$ for $n > 1$
- List the first few Fibonacci numbers
We are given the first two Fibonacci numbers, $F(0) = 0$ and $F(1) = 1$.
- Calculate subsequent Fibonacci numbers
Using the recursive formula, we can calculate the next Fibonacci numbers until we reach the 13th number (which is actually the 12th index since we start from index 0).
$F(2) = F(1) + F(0) = 1 + 0 = 1$ $F(3) = F(2) + F(1) = 1 + 1 = 2$ $F(4) = F(3) + F(2) = 2 + 1 = 3$ $F(5) = F(4) + F(3) = 3 + 2 = 5$ $F(6) = F(5) + F(4) = 5 + 3 = 8$ $F(7) = F(6) + F(5) = 8 + 5 = 13$ $F(8) = F(7) + F(6) = 13 + 8 = 21$ $F(9) = F(8) + F(7) = 21 + 13 = 34$ $F(10) = F(9) + F(8) = 34 + 21 = 55$ $F(11) = F(10) + F(9) = 55 + 34 = 89$ $F(12) = F(11) + F(10) = 89 + 55 = 144$
- Identify the 13th Fibonacci number
The 13th number in the sequence corresponds to $F(12)$, which we calculated to be 144.
144
More Information
The Fibonacci sequence appears in various natural phenomena, such as the arrangement of leaves on a stem, the patterns of flower petals, and the spirals of shells.
Tips
A common mistake is to start the sequence with 1 and 1 instead of 0 and 1, which will lead to an incorrect result. Another mistake is miscounting the terms, for example, stopping at $F(12)$ but calling it the 12th term instead of the 13th. It's also possible to make arithmetic errors when adding the numbers.
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