Math Problem Statement
Fibonacci sequence F(23)=?
Solution
To calculate the 23rd Fibonacci number, , we follow the Fibonacci sequence definition, where each number is the sum of the two preceding ones:
with the initial conditions and .
Calculating step-by-step:
&F(0) = 0, \\ &F(1) = 1, \\ &F(2) = 1, \\ &F(3) = 2, \\ &F(4) = 3, \\ &F(5) = 5, \\ &F(6) = 8, \\ &F(7) = 13, \\ &F(8) = 21, \\ &F(9) = 34, \\ &F(10) = 55, \\ &F(11) = 89, \\ &F(12) = 144, \\ &F(13) = 233, \\ &F(14) = 377, \\ &F(15) = 610, \\ &F(16) = 987, \\ &F(17) = 1597, \\ &F(18) = 2584, \\ &F(19) = 4181, \\ &F(20) = 6765, \\ &F(21) = 10946, \\ &F(22) = 17711, \\ &F(23) = 28657. \end{aligned}$$ Thus, $$F(23) = 28,657$$. Would you like further details or have any questions? --- Here are five related questions to expand on this topic: 1. How do you derive the closed-form expression for the Fibonacci sequence? 2. What is the significance of the golden ratio in the Fibonacci sequence? 3. Can you explain the relationship between Fibonacci numbers and Pascal's Triangle? 4. How do Fibonacci numbers appear in nature? 5. What are some applications of the Fibonacci sequence in computer science? **Tip:** The Fibonacci sequence grows exponentially, and you can use matrix exponentiation to compute large Fibonacci numbers efficiently.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Fibonacci Sequence
Recursive Sequences
Formulas
Fibonacci sequence formula: F(n) = F(n-1) + F(n-2)
Theorems
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Suitable Grade Level
Grades 9-12