Difference between revisions of "Fibonacci sequence"
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==Problems== | ==Problems== | ||
=== Introductory === | === Introductory === | ||
| + | # The Fibonacci sequence <math>1,1,2,3,5,8,13,21,\ldots </math> starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten [[digit]]s is the last to appear in the units position of a number in the Fibonacci sequence?<br><br><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 } </math><div style="text-align:right">([[2000 AMC 12 Problems/Problem 4|2000 AMC 12, Problem 4]])</div> | ||
=== Intermediate === | === Intermediate === | ||
# Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. <div style="text-align:right">(Manhattan Mathematical Olympiad 2004)</div> | # Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. <div style="text-align:right">(Manhattan Mathematical Olympiad 2004)</div> | ||
# A [[fair]] coin is to be tossed <math>10_{}^{}</math> times. Let <math>i/j^{}_{}</math>, in lowest terms, be the [[probability]] that heads never occur on consecutive tosses. Find <math>i+j_{}^{}</math>. <div style="text-align:right">([[1990 AIME Problems/Problem 9|1990 AIME, Problem 9]])</div> | # A [[fair]] coin is to be tossed <math>10_{}^{}</math> times. Let <math>i/j^{}_{}</math>, in lowest terms, be the [[probability]] that heads never occur on consecutive tosses. Find <math>i+j_{}^{}</math>. <div style="text-align:right">([[1990 AIME Problems/Problem 9|1990 AIME, Problem 9]])</div> | ||
=== Olympiad === | === Olympiad === | ||
| + | # Determine the maximum value of <math> \displaystyle m^2 + n^2 </math>, where <math> \displaystyle m </math> and <math> \displaystyle n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math> \displaystyle ( n^2 - mn - m^2 )^2 = 1 </math>. <div style="text-align:right">([[1981 IMO Problems/Problem 3|1981 IMO, Problem 3]])</div> | ||
==See also== | ==See also== | ||
Revision as of 17:15, 4 March 2007
The Fibonacci sequence is a sequence of integers in which the first and second term are both equal to 1, and each subsequent term is the sum of the two preceding it. The first few terms are 
.
The Fibonacci sequence can be written recursively as 
.
Contents
Phi
Ratios between successive terms, 
, 
, 
, 
, 
, tend towards the limit phi.
Binet's formula
Binet's formula is an explicit formula used to find any nth term.
It is 
.
Problems
Introductory
- The Fibonacci sequence
starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten 
Intermediate
- Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. (Manhattan Mathematical Olympiad 2004)
- A fair coin is to be tossed
times. Let 
, in lowest terms, be the probability that heads never occur on consecutive tosses. Find 
.
times. Let 
, in lowest terms, be the 
. Olympiad
- Determine the maximum value of
, where 
and 
are integers satisfying 
and 
.
, where 
and 
are integers satisfying 
and 
. See also
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