What is the Fibonacci sequence?

The Fibonacci sequence is a series where each number is the sum of the two preceding ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... It starts with F(1)=1, F(2)=1, and continues with F(n) = F(n-1) + F(n-2).

What is the golden ratio and how is it related to Fibonacci?

The golden ratio (phi) is approximately 1.618033988749895. As Fibonacci numbers get larger, the ratio of consecutive Fibonacci numbers approaches the golden ratio: F(n)/F(n-1) approaches phi as n approaches infinity.

How can I check if a number is a Fibonacci number?

A number N is a Fibonacci number if and only if one of (5N² + 4) or (5N² - 4) is a perfect square. This calculator uses this mathematical property to verify Fibonacci numbers.

What is Binet's formula?

Binet's formula calculates the nth Fibonacci number directly: F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1+sqrt(5))/2 (golden ratio) and psi = (1-sqrt(5))/2. This gives exact results for smaller n values.

What are some applications of Fibonacci numbers?

Fibonacci numbers appear in nature (flower petals, pinecones, shells), art and architecture (golden ratio proportions), computer science (algorithms, data structures), financial markets (Fibonacci retracement), and mathematics (number theory).

What is the sum of the first n Fibonacci numbers?

The sum of the first n Fibonacci numbers equals F(n+2) - 1. For example, the sum of the first 5 Fibonacci numbers (1+1+2+3+5=12) equals F(7)-1 = 13-1 = 12.

Fibonacci Calculator

Calculate Fibonacci numbers, generate sequences, check if a number is Fibonacci, and explore the golden ratio. See step-by-step solutions with multiple calculation methods.

star 4.8 (3,102 ratings)

calendar_today Updated:

pattern Fibonacci Calculator

Calculation Type

Position (n)

1-based position (max 78 for safe integers)

Golden Ratio (φ)

1.6180339887...

F(n)/F(n-1) approaches φ as n increases

The Pattern

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

Where F(1) = 1, F(2) = 1

calculate Result

F(10)

F(n)/F(n-1)

1.6176...

Approaching golden ratio (φ)

Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Did You Know?

• Fibonacci appears in nature: flower petals, pinecones, shells

• Used in art, architecture, and financial trading

Input Values

Calculation Type

expand_more

n (Position or Count)

Number to Check

Show Steps

Yes

lightbulb Tips

•F(n) = F(n-1) + F(n-2)
•F(1) = F(2) = 1
•φ ≈ 1.618 (golden ratio)
•Sum = F(n+2) - 1

pattern Reference

Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

Key Values

F(10) = 55F(15) = 610 F(20) = 6,765F(30) = 832,040

Golden Ratio

φ = (1+√5)/2 ≈ 1.618034

How to Use This Calculator

calculate

Choose Calculation

Find nth term, generate sequence, check if Fibonacci, or calculate sum.

edit

Enter Position/Number

Input the position n or number to check.

visibility

View Results

See the Fibonacci number, sequence, or verification.

The Formula

Fibonacci numbers form a sequence where each term is the sum of the two preceding terms, starting with 1, 1.

F(n) = F(n-1) + F(n-2), where F(1) = F(2) = 1

lightbulb Variables Explained

F(n) The nth Fibonacci number
φ Golden ratio ≈ 1.618033988749895
Binet F(n) = (φⁿ - ψⁿ)/√5

tips_and_updates Pro Tips

The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...

Each number is the sum of the two before it: F(n) = F(n-1) + F(n-2)

The ratio of consecutive Fibonacci numbers approaches the golden ratio (φ ≈ 1.618)

Fibonacci numbers appear in nature: flower petals, pinecones, shells

A number is Fibonacci if 5n² + 4 or 5n² - 4 is a perfect square

The sum of first n Fibonacci numbers equals F(n+2) - 1

Fibonacci Number Calculator

Calculate Fibonacci numbers, generate sequences, check if a number is Fibonacci, and explore the golden ratio connection. See step-by-step solutions.

The Fibonacci Sequence

Starting with 1, 1, each subsequent number is the sum of the previous two. This creates the famous sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

Golden Ratio Connection

As Fibonacci numbers grow, the ratio between consecutive terms approaches the golden ratio φ ≈ 1.618. This ratio appears throughout nature, art, and architecture.