🎓 Mathematics • Sequences & Series
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Fibonacci Calculator
Generates Fibonacci sequence terms up to n=200 using arbitrary-precision bigints and displays sequence history.
Arithmetic Progression Parameters
AP Quick Presets:
Progressive Solution (n = 10)
10-th Term (a_10)
48
aₙ = a₁ + (n - 1)dSum of First 10 Terms (S_10)
255
Sₙ = (n/2)(a₁ + aₙ)Sequence Terms:
a_1:3
a_2:8
a_3:13
a_4:18
a_5:23
a_6:28
a_7:33
a_8:38
a_9:43
a_10:48
Mathematical Derivation & Formulas:
1. N-th Term Formula: aₙ = a₁ + (n - 1)d
= 3 + (10 - 1) × 5 = 3 + 45 = 48
2. Sum Formula: Sₙ = (n / 2) × (a₁ + aₙ)
= (10 / 2) × (3 + 48) = 5 × 51 = 255
What is the Fibonacci Sequence?
The Fibonacci sequence begins with 0 and 1, where each subsequent number is the sum of the preceding two.
The ratio of successive terms converges to the Golden Ratio φ ≈ 1.6180339887...
Formula & Step-by-Step Calculation
F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂
Recursive second-order linear recurrence sequence.
Worked Step-by-Step Examples
Example 1
F₁₀
Solution: 55
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Common Real-World & Academic Use Cases
- ✓ Algorithm recursion study
- ✓ Financial Fibonacci retracement trading
- ✓ Botanical phyllotaxis spiral modeling
How to Use the Fibonacci Calculator
1
Input n
Enter term index n.
2
Read Fibonacci Number
Inspect exact integer and sequence list.
Frequently Asked Questions
Q: What is the Golden Ratio?
φ = (1 + √5) / 2 ≈ 1.61803398875.