Fibonacci Sequence Calculator
Calculate the nth Fibonacci number and generate the sequence. F(n) = F(n-1) + F(n-2). Explore the golden ratio φ = 1.61803 and Fibonacci patterns in nature.
🔢 Number Theory📐 F(n) = F(n-1) + F(n-2)🔢 Math
Find F(n)
Generate first k terms
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Fibonacci Sequence Calculator | — | F(n) = F(n-1) + F(n-2) | integer |
Step-by-Step Examples
Example 1
F(10)
F(10)=?
- 0,1,1,2,3,5,8,13,21,34,55
- F(10)=55; ratio 55/34≈1.618
✓ 55
Example 2
Golden ratio
Consecutive ratios approach φ
- F(20)/F(19)=6765/4181=1.61803
- φ=(1+√5)/2=1.61803398...
✓ → 1.61803
Example 3
First 12
Generate
- 0,1,1,2,3,5,8,13,21,34,55,89
✓ F(12)=144
Real-World Applications
Nature
Sunflower seeds, pinecones, flower petals.
Finance
Fibonacci retracement: 23.6%, 38.2%, 61.8%.
CS
Fibonacci heap, dynamic programming.
Art
Golden ratio in art and architecture.
Common Mistakes to Avoid
⚠️
Index confusion
F(0)=0 or F(1)=1? Clarify the convention.
⚠️
Naive recursion
O(2^n) — use iteration or memoization for speed.
⚠️
φ is irrational
Ratio approaches but never exactly equals φ.
Frequently Asked Questions
What is Fibonacci? ▾
0,1,1,2,3,5,8,13... Each term=sum of two before.
Golden ratio? ▾
φ=(1+√5)/2≈1.618. Ratio converges to φ.
Binet's formula? ▾
F(n)=(φ^n-ψ^n)/√5, exact closed form.
In nature? ▾
Spirals in sunflowers, nautilus shells, leaf angles.
Golden spiral? ▾
Quarter circles in Fibonacci squares form the spiral.
Related Math Calculators
Formula Explorer connections
Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.