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Browse High School Fibonacci Sequence/Golden Ratio

Stars indicate particularly interesting answers or good places to begin browsing.

The Golden Ratio [02/23/1998]
I know that the limit of the ratios of the Fibonnaci sequence is the golden mean, but I would like to see a proof.

Golden Ratio and the Sine of 18 [04/17/2001]
2*sin(18) + 1 is equal to the golden ratio. Is there any significance to this?

Golden Rectangle History [9/7/1996]
What is the history of the golden rectangle? When was it discovered and by whom?

Golden Triangle: An Isosceles Triangle [01/23/2001]
What is the Golden Triangle?

Golden Triangle: What is It? [09/19/1999]
What is a Golden Triangle?

Information on the Golden Ratio [3/30/1996]
Would it be possible for you to send me some information on the Golden Ratio?

Last Four Digits of the Fibonacci Numbers [05/06/2001]
Show that there is a number ending with four zeros in the Fibonacci sequence; prove that the Fibonacci sequence has a cycle for the last four digits with a length of 15,000.

The Law of Margins [9/2/1996]
How do I figure out the margins of my mat using the Golden Ratio?

Lucas Numbers [11/22/1996]
Is 2 a Lucas number?

Multiplying Mice [07/23/1997]
Baby mice can breed when they are 6 weeks old and the babies are born after 3 weeks. If each mother mouse has only one litter and all the litters have 8 babies, half males and half females, how many mice will you have 18 weeks from today?

Non-Recursive Formula [06/05/2001]
I want to know the non-recursive formula of the nth number in the general Fibonacci sequence...

Numbers in the Fibonacci Sequence [07/19/2001]
How can I show that there is a number in the Fibonacci sequence that ends in 999999999999 ? For what numbers n is there a number in the Fibonacci sequence that ends in n of 9 ?

The Origin of Lucas Numbers [10/08/1998]
I need help with Lucas Numbers - how and why they were created.

Phi [05/18/1997]
What is phi?

phi vs. Phi - a Coincidence? [10/16/2001]
Ancient and modern architecture reflect the 'golden ratio' (1.618. length to width) and this number is remarkably close to phi (.618...) seen in nature for leaf dispersions, etc. Is this just a coincidence?

Primes in Fibonacci [07/11/1999]
Is there any prime number pattern in the Fibonacci sequence?

Proof of Convergence [09/29/2000]
Why does the ratio F(n+1)/F(n) for the Fibonacci numbers converge to the golden ratio?

Recursive vs. Explicit Formulas [01/02/1997]
What is the difference between explicit and recursive formulas?

Silver Means [01/26/2007]
I've found a number that is similar to the golden mean--it's derived and behaves in much the same way. Is this a special number also or is there a simple mathematical explanation?

Simple Number Pair Series Yields Surprising Ratio ... Why? [12/31/2009]
An enthusiast wonders about the curious ratio that emerges from a simple pattern for generating number pairs. Doctor Rick builds an algebraic argument for why its phi-like recursive relationship approaches the square root of 2.

Sum of Distinct Fibonacci Numbers [05/06/2001]
How do you show that every positive integer is a sum of distinct terms of the Fibonacci sequence?

Sum of Fibonacci Series [05/23/2000]
Is there a formula for the sum of the first n numbers in the Fibonacci sequence?

Tribonacci Numbers [11/11/2000]
Is there an implicit formula to calculate the nth Tribonacci number? Also, is there a formula to find the sum of the first n Tribonacci numbers?

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