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Browse High School Fibonacci Sequence/Golden Ratio

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



Appearances of the Golden Number [06/03/1998]
Why does the irrational number phi = (1 + sqrt(5))/2 appear in so many biological and non-biological applications?

Calculating the Fibonacci Sequence [11/28/1996]
Is there a formula to calculate the nth Fibonacci number?

Congruum Problem [04/04/2002]
I have found a reference to Fibonacci and his congruum problem. But something has me stumped...

Fibonacci sequence in nature, Golden Mean, Golden Ratio [12/6/1994]
I need examples of where the Fibonacci sequence is found in nature and how it relates to the Golden Mean.

Fibonacci Series [11/7/1994]
I was helping an Algebra student with a "bonus" problem recently. It asked something about drawing a spiral using the Fibonacci series. What is this series? Does it draw a spiral?

Golden Ratio [01/03/1998]
Do you have any topics that I can use in my term paper about the golden ratio?

The Relation of the Golden Ratio and the Fibonacci Series [1/28/1996]
I am trying to figure out how the Golden Ratio and the Fibonacci series are related.

Why Is Math Important? [01/02/2002]
Some of it we don't need to know in the real world, so why do they teach us things we won't need to know?

The 1000th Fibonacci Number [09/25/1998]
How can you find the 1000th Fibonacci number, both approximate and exact?

500th Fibonacci Number [04/16/1997]
How do you find the 500th term of the Fibonacci sequence?

Binet's Formula and Induction [11/28/2001]
What is induction? Can you prove Binet's formula by induction?

Consecutive Fibonacci Numbers Relatively Prime [11/17/2001]
Prove two consecutive Fibonacci numbers are relatively prime.

Constructing the Golden Ratio Using a Compass [12/30/2003]
I am writing a paper on the Golden Ratio and want to include a section on how to construct the ratio on a line using a compass. Could you please explain the construction?

Expressing the Golden Ratio as a Decimal [03/04/1998]
Using Newton's method to express the value l+sqrt(5)/2 as a decimal.

Fibonacci [01/31/1997]
Where can I learn more about Leonard of Pisa?

Fibonacci Addition Trick [10/10/2002]
Given 10 numbers in a Fibonacci sequence, why does multiplying the seventh number by 11 give the sum of all 10 numbers?

Fibonacci Algorithm and Egyptian Fractions [04/25/1999]
How long an Egyptian fraction can you write?

A Fibonacci and Lucas Number Relation [08/24/1998]
Prove that F(2n) = F(n) * L(n) where F(x) is the xth Fibonacci number and L(y) is the yth Lucas number.

Fibonacci Formula Inductive Proof [11/05/1997]
I must prove by induction that F(n) = (PHI^n - (1 - PHI)^n) / sqrt5...

Fibonacci-GCD Proof [11/20/2002]
Can you help me prove that fib(gcd(m, n)) = gcd(fib(m), fib(n)) ?

Fibonacci Identity [12/10/2001]
I am trying to create an inductive proof for the particular identity of Fibonacci numbers that: F(n-1) * F(n+1) = (-1)^n + (Fn)^2.

A Fibonacci Jigsaw Puzzle [04/29/1999]
Why is the area of our rectangle, formed from a square, 65 when the square's area was 64?

Fibonacci Numbers [9/25/1995]
What is the rule of Fibonacci numbers in math and life?

Fibonacci Numbers in Pascal's Triangle [04/03/2003]
How do you find the Fibonacci numbers in Pascal's Triangle?

Fibonacci Proof [01/29/2001]
This proof is giving me major problems: F(2n) = (F(n))^2 + (F(n-1))^2. ...

A Fibonacci Proof by Induction [06/05/1998]
Let u_1, u_2, ... be the Fibonacci sequence. Prove by induction...

Fibonacci Ratio, Golden Ratio, and 8:5 [06/01/1997]
What is the connection between the ratio 8:5, the Fibonacci sequence, and the Golden Ratio?

Fibonacci Recurred, Composed, Summed [02/18/2011]
Given g(n) = 2n + 1, how do you show that sum{f(4n + 3)} = sum{f(g(g(n)))}? Doctor Vogler shows how to begin by using an explicit Fibonacci formula and sum of a geometric series.

Fibonacci Riddle [11/21/2001]
We can cut an 8x8 square with an area of 64 into four pieces and reassemble to get a 5x13 rectangle with an area of 65. Where does the extra 1x1 square come from?

Fibonacci sequence [1/28/1996]
What is the explicit formula for the Fibonacci numbers?

Fibonacci Sequence [01/29/2001]
Is there a formula for the n-th Fibonacci number?

Fibonacci Sequence [06/18/2001]
I need help understanding what the Fibonacci sequence really is.

Fibonacci Sequence and Grapefruit [12/6/1995]
How can the Fibonacci recursive sequence be related to or used to explain the appearance of the sections of a grapefruit sliced in half?

Fibonacci Sequence - An Example [05/12/1999]
Glass plates and reflections.

Fibonacci sequence, Golden Ratio, Golden Rectangle, Industry [11/16/1994]
Do you know of any industries that purposefully employ the Fibonacci Sequence in their product design?

Fibonacci Sequence in Nature [12/08/1997]
I need examples of how the Fibonacci sequence is used in nature.

Fibonacci Sequence Property [11/29/2001]
I have to prove that in the Fibonacci sequence, F(k) is a divisor of F(nk), where n is a natural number (so, F(nk) = A*F(k) where A is a natural number).

Fibonacci Sequences [01/08/1998]
Please help me with a proof.

Fibonacci Trick [06/01/1999]
The Fibonacci trick states that F(n+1) * F(n-1) - [F(n)]^2 = (-1)^n. Why does this work?

Formula for the Fibonacci Sequence [4/8/1996]
What is the implicit formula for the Fibonacci sequence?

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