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 TOPICS This page:   Fibonacci sequence,   golden ratio    Search   Dr. Math See also the Dr. Math FAQ:   golden ratio,   Fibonacci sequence Internet Library:   golden ratio/   Fibonacci HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry 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? Page:  1  2 [next>]

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