Concepts · 3 min read
Fibonacci Sequence and the Golden Ratio, Computed
The sequence
Start with 0 and 1 and keep adding.
| n | F(n) | n | F(n) |
|---|---|---|---|
| 0 | 0 | 10 | 55 |
| 1 | 1 | 11 | 89 |
| 2 | 1 | 12 | 144 |
| 3 | 2 | 13 | 233 |
| 4 | 3 | 14 | 377 |
| 5 | 5 | 15 | 610 |
| 6 | 8 | 16 | 987 |
| 7 | 13 | 17 | 1597 |
| 8 | 21 | 18 | 2584 |
| 9 | 34 | 19 | 4181 |
Ratios converge to φ
| n | F(n+1) ÷ F(n) |
|---|---|
| 5 | 1.6 |
| 8 | 1.619047619 |
| 10 | 1.618181818 |
| 15 | 1.618032787 |
| 20 | 1.618033999 |
| 25 | 1.618033989 |
φ itself is 1.618033989, and it satisfies φ² = φ + 1 (check: φ² = 2.618033989). The convergence is very fast.
Binet’s formula
F(n) = (φⁿ − ψⁿ)/√5 where ψ = (1 − √5)/2. For n = 10: 55, matching F(10) = 55.
φ^10 ≈ 122.99 (the dominant term)
Step-by-step working 3 steps
- 1Write it as a power
1.618034^10
- 2Repeated multiplication
1.618034 is multiplied by itself 10 times.
- 3Result
1.618034^10 = 122.99186938
Connections
- Sums: F(1) + … + F(n) = F(n+2) − 1.
- Fibonacci numbers appear as sums along the diagonals of Pascal’s triangle.
- Consecutive Fibonacci numbers are coprime.
Check coprimality with the LCM and GCF calculator: LCM 33552 · GCF 1.
Frequently asked questions
What is the golden ratio?
About 1.6180339887, the positive solution of x² = x + 1.
Is the sequence infinite?
Yes, and it grows exponentially: F(30) = 832,040.
Keep reading
What Is Euler's Number e? Compound Growth and Series
e ≈ 2.718281828 explained through compound interest, the limit (1 + 1/n)ⁿ and the factorial series, with computed tables.
ConceptsPascal's Triangle and the Binomial Theorem Explained
Build Pascal's triangle, see how each entry is a combination nCr, and expand (a + b)ⁿ using its rows.
ConceptsLogarithms Explained Simply: What They Are and Why They Matter
Logarithms as “the power you need”, with tables for base 10 and base 2, the product rule, and real uses like pH and the Richter scale.
ConceptsCalculus Basics: Derivatives and Integrals With Numbers
A gentle introduction to derivatives as slopes and integrals as areas, with numerical tables showing slopes and sums converging to exact answers.