Math Problem Statement
The Fibonacci sequence can be used to approximate the Golden Ratio. Starting
with the sequence 1, 1, 2, 3, 5, 8, 13, find the ratio between the 8th and 7th
terms. What does this ratio approach as the terms increase?
Solution
The Fibonacci sequence starts as:
The ratio between the 8th and 7th terms of the sequence is:
As the terms of the Fibonacci sequence increase, the ratio between consecutive terms approaches the Golden Ratio (denoted by ), which is approximately:
This means that, as you calculate the ratio of consecutive Fibonacci numbers further into the sequence, the ratio becomes closer to 1.618.
Would you like more details or have any questions?
Here are 5 related questions:
- What is the exact mathematical expression for the Golden Ratio?
- How is the Fibonacci sequence connected to nature and art?
- Can the Golden Ratio be derived using algebra or geometry?
- What other mathematical sequences exhibit similar ratio patterns?
- How can the Golden Ratio be used in real-world applications like design and architecture?
Tip: To observe the approach to the Golden Ratio, calculate the ratio for larger terms in the Fibonacci sequence (e.g., 21st and 20th terms).
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Fibonacci Sequence
Ratios
Golden Ratio
Formulas
Ratio between consecutive Fibonacci numbers: F(n+1)/F(n)
Golden Ratio formula: φ = (1 + √5) / 2
Theorems
Golden Ratio Theorem
Suitable Grade Level
Grades 8-10
Related Recommendation
Exploring Fibonacci Sequence and Golden Ratio in Spirals
Fibonacci Sequence: Formula, Applications, and Key Properties
Exploring Powers of the Golden Ratio's Conjugate (0.6180339887)
Understanding Fibonacci Sequence: Limit of Fn/Fn+1 as n Grows
Understanding the Fibonacci Sequence with Recursive Formula