To what value will Fn/Fn+1 approaches to as n gets bigger?

As \( n \) gets larger, the ratio \( \frac{F_n}{F_{n+1}} \) of consecutive Fibonacci numbers approaches the value of \( \frac{1}{\phi} \), where \( \phi \) (the golden ratio) is approximately \( 1.618033988749895 \). 

The golden ratio \( \phi \) is defined as:

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

Thus, the value \( \frac{F_n}{F_{n+1}} \) approaches is:

\[ \frac{1}{\phi} = \frac{2}{1 + \sqrt{5}} = \phi - 1 \approx 0.618033988749895 \]

This relationship can be derived from the properties of the Fibonacci sequence and the characteristic equation associated with it.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the characteristic equation of the Fibonacci sequence?
2. How can the golden ratio be derived from the quadratic equation?
3. What is the significance of the golden ratio in mathematics and nature?
4. Can the relationship \( \frac{F_n}{F_{n+1}} \approx 0.618 \) be proved using mathematical induction?
5. How does the ratio \( \frac{F_{n+1}}{F_n} \) relate to the golden ratio?

**Tip:** The golden ratio appears in many different areas of mathematics and nature, from the proportions of geometric shapes to the growth patterns of plants. Exploring these connections can provide a deeper appreciation of this fascinating number.

Explore the limit of the ratio Fn/Fn+1 as n grows larger in the Fibonacci sequence. Learn how this ratio approaches the value of 1/φ, where φ is the golden ratio (approximately 1.618033988749895). Dive into mathematical proofs and applications of this theorem. Suitable for advanced high school students.

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