Fibonacci Sequence/Golden Ratio

• Appearances of the Golden Number [Matias, 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 [Suhendra, 11/28/1996]
  Is there a formula to calculate the nth Fibonacci number?

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

• Fibonacci Series [Jones, 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?

• Fibonacci sequence in nature, Golden Mean, Golden Ratio [Anderson, 12/6/1994]
  For my Pre-Calculus class I need examples of where the Fibonacci sequence is found in nature and how it relates to the Golden Mean.

• Golden Ratio [Park, 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 [Mixecoxt, 1/28/1996]
  I am trying to figure out how the Golden Ratio and the Fibonacci series are related.

• Why Is Math Important? [Amanda, 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?

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

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

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

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

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

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

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

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

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

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

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

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

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

• Multiplying Mice [Slavic, 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?

• Brick Patterns [Jessi, 09/17/1997]
  How many different ways can 1x2 ft. bricks be arranged to make a 20' long path?

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

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

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

• The Golden Ratio [Miller, 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.

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

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

• A Fibonacci and Lucas Number Relation [Ng, 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.

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

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

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

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

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

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

• Golden Triangle [Byerly, 09/19/1999]
  What is a Golden Triangle?

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

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

• Tribonacci Numbers [aznx, 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?

• Golden Triangle [Connors, 01/23/2001]
  What is the Golden Triangle?

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

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

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

• Last Four Digits of the Fibonacci Numbers [Sam, 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.

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

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

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

• phi vs. Phi - a Coincidence? [Cotton, 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?

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

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

• Fibonacci Sequence Property [Enrique, 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 Identity [Doug, 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.

• Numbers in the Fibonacci Sequence [Dinu, 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 ?