Question
What is Fibonacci sequence and it's example
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GPT-4o mini
The Fibonacci sequence is a series of numbers in which each number (after the first two) is the sum of the two preceding ones. The sequence typically starts with 0 and 1. Mathematically, it can be defined by the recurrence relation:
\[ F(n) = F(n-1) + F(n-2) \]
with initial conditions
\[ F(0) = 0, \quad F(1) = 1. \]
The beginning of the Fibonacci sequence is:
- \( F(0) = 0 \)
- \( F(1) = 1 \)
- \( F(2) = F(1) + F(0) = 1 + 0 = 1 \)
- \( F(3) = F(2) + F(1) = 1 + 1 = 2 \)
- \( F(4) = F(3) + F(2) = 2 + 1 = 3 \)
- \( F(5) = F(4) + F(3) = 3 + 2 = 5 \)
- \( F(6) = F(5) + F(4) = 5 + 3 = 8 \)
- \( F(7) = F(6) + F(5) = 8 + 5 = 13 \)
- \( F(8) = F(7) + F(6) = 13 + 8 = 21 \)
- \( F(9) = F(8) + F(7) = 21 + 13 = 34 \)
- \( F(10) = F(9) + F(8) = 34 + 21 = 55 \)
So the beginning of the Fibonacci sequence is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
The Fibonacci sequence appears in various areas of mathematics and nature, often associated with properties related to growth patterns, phyllotaxis in plants, and the arrangement of leaves or petals. It’s also related to the golden ratio, as the ratio of successive Fibonacci numbers approximates the golden ratio as the numbers increase.
\[ F(n) = F(n-1) + F(n-2) \]
with initial conditions
\[ F(0) = 0, \quad F(1) = 1. \]
The beginning of the Fibonacci sequence is:
- \( F(0) = 0 \)
- \( F(1) = 1 \)
- \( F(2) = F(1) + F(0) = 1 + 0 = 1 \)
- \( F(3) = F(2) + F(1) = 1 + 1 = 2 \)
- \( F(4) = F(3) + F(2) = 2 + 1 = 3 \)
- \( F(5) = F(4) + F(3) = 3 + 2 = 5 \)
- \( F(6) = F(5) + F(4) = 5 + 3 = 8 \)
- \( F(7) = F(6) + F(5) = 8 + 5 = 13 \)
- \( F(8) = F(7) + F(6) = 13 + 8 = 21 \)
- \( F(9) = F(8) + F(7) = 21 + 13 = 34 \)
- \( F(10) = F(9) + F(8) = 34 + 21 = 55 \)
So the beginning of the Fibonacci sequence is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
The Fibonacci sequence appears in various areas of mathematics and nature, often associated with properties related to growth patterns, phyllotaxis in plants, and the arrangement of leaves or petals. It’s also related to the golden ratio, as the ratio of successive Fibonacci numbers approximates the golden ratio as the numbers increase.