What Happens When You Randomize the Fibonacci Sequence?

TL;DR

Randomizing the Fibonacci sequence by using a coin flip to determine addition or subtraction results in unpredictable sequences that still exhibit a long-term growth. The growth rate of these random sequences approaches approximately 1.13198824, allowing predictions of sequential values despite their chaotic nature.

Transcript

We're gonna do randomised Fibonacci sequences. So, we're gonna take a Fibonacci sequence, we're gonna randomise it ....with this coin. Now, I think we should first do a recap of Fibonacci sequences first, just to see what the original Fibonacci sequence is. So, your Fibonacci sequence starts with a 1, and a 1, and then the next term is the sum of t... Read More

Key Insights

• 🤑 Fibonacci sequences follow a specific pattern, starting with two ones and each subsequent number being the sum of the previous two.
• #️⃣ Dividing two consecutive Fibonacci numbers tends to the golden ratio, a special number denoted by the symbol Phi (1.618033988...).
• #️⃣ The golden ratio can be used to estimate large Fibonacci numbers when the exact numbers are unknown.
• 🥳 Randomizing the Fibonacci sequence creates unpredictable sequences, and their long-term growth is determined by a constant value similar to the golden ratio.
• ☠️ The growth rate of random Fibonacci sequences was officially calculated in 1999 to be approximately 1.13198824, with more digits being discovered since then.
• 🥳 Applications of Fibonacci sequences and the golden ratio can be found in various fields, including mathematics, art, and nature.
• 🚨 The unpredictable nature of random Fibonacci sequences makes it unlikely for traditional Fibonacci patterns or specific patterns to emerge.

Questions & Answers

Q: What is a Fibonacci sequence?

A Fibonacci sequence starts with two ones, and each subsequent number is the sum of the previous two numbers.

Q: What is the golden ratio?

The golden ratio is a special number, approximately 1.618033988..., that can be used to estimate large Fibonacci numbers and has various applications in mathematics and art.

Q: How can the golden ratio be used to estimate Fibonacci numbers?

By multiplying the golden ratio by itself a certain number of times (equal to the position of the desired Fibonacci number), an estimate of the large Fibonacci number can be obtained.

Q: How does randomizing the Fibonacci sequence work?

Randomizing the Fibonacci sequence involves flipping a coin to determine whether to add or subtract the previous two numbers, creating unpredictable sequences that still grow.

Summary & Key Takeaways

• Fibonacci sequences start with two ones and each subsequent number is the sum of the previous two numbers.

• The golden ratio, 1.618033988..., is a special number related to Fibonacci sequences and can be used to estimate large Fibonacci numbers.

• Randomizing the Fibonacci sequence by flipping a coin to determine whether to add or subtract the previous two numbers creates unpredictable sequences that still exhibit growth.