The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It starts with $F(0) = 0$, $F(1) = 1$, and every subsequent number is defined by the recurrence relation:

$F(n) = F(n-1) + F(n-2)$

We are tasked with finding the 55th Fibonacci number, denoted as $F(55)$.

Steps:

1. Base Case:

   • $F(0) = 0$
   • $F(1) = 1$

2. Recursive Formula:

   • To compute the Fibonacci numbers, use the formula $F(n) = F(n-1) + F(n-2)$.

3. Calculation Using an Iterative Approach: We can compute the Fibonacci numbers iteratively from $F(2)$ to $F(55)$ using the recurrence relation.

   • $F(2) = F(1) + F(0) = 1 + 0 = 1$
   • $F(3) = F(2) + F(1) = 1 + 1 = 2$
   • $F(4) = F(3) + F(2) = 2 + 1 = 3$
   • Continue this process up to $F(55)$.

Direct Result:

After performing the iterative calculations, the 55th Fibonacci number is:

$F(55) = 139,583,862,445$

Thus, the 55th Fibonacci number is $139,583,862,445$.

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Would you like to explore more about the Fibonacci sequence? Here are some related questions:

1. How do Fibonacci numbers relate to the golden spiral?
2. Can the Fibonacci numbers be expressed using a closed-form formula (Binet’s formula)?
3. What is the time complexity of calculating Fibonacci numbers using recursion vs iteration?
4. How are Fibonacci numbers used in computer algorithms?
5. How do Fibonacci numbers appear in nature (e.g., sunflower patterns)?

Tip: When calculating large Fibonacci numbers, it's efficient to use an iterative method instead of recursion to avoid excessive computation and stack overflow issues.