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?

The Fibonacci sequence starts as: 

\[ 1, 1, 2, 3, 5, 8, 13, 21, \dots \]

The ratio between the 8th and 7th terms of the sequence is:

\[
\frac{\text{8th term}}{\text{7th term}} = \frac{21}{13} \approx 1.615
\]

As the terms of the Fibonacci sequence increase, the ratio between consecutive terms approaches the **Golden Ratio** (denoted by \( \phi \)), which is approximately:

\[
\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618
\]

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:
1. What is the exact mathematical expression for the Golden Ratio?
2. How is the Fibonacci sequence connected to nature and art?
3. Can the Golden Ratio be derived using algebra or geometry?
4. What other mathematical sequences exhibit similar ratio patterns?
5. 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).

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?

Explore how the Fibonacci sequence can be used to approximate the Golden Ratio. Starting with the sequence 1, 1, 2, 3, 5, 8, 13, discover the ratio between the 8th and 7th terms and how this ratio approaches the Golden Ratio of approximately 1.618. Learn key mathematical concepts, including the Fibonacci sequence and its relationship to the Golden Ratio, suitable for students in Grades 8-10.

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