Difference between revisions of "Fibonacci sequence"
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=== 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> | # 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> | ||
+ | # Find <math>gcd(F_n,F_{n+1})</math>. | ||
+ | |||
=== 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> | ||
# Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first [[negative]] term encounted. What positive integer <math>x</math> produces a sequence of maximum length? <div style="text-align:right">([[1998 AIME Problems/Problem 8|1998 AIME, Problem 8]])</div> | # Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first [[negative]] term encounted. What positive integer <math>x</math> produces a sequence of maximum length? <div style="text-align:right">([[1998 AIME Problems/Problem 8|1998 AIME, Problem 8]])</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> | ||
+ | #Find <math>a</math> if <math>a</math> and <math>b</math> are [[integer]]s such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. <div style="text-align:right">([[1998 AIME Problems/Problem 13|1998 AIME, Problem 13]])</div> | ||
+ | |||
=== Olympiad === | === Olympiad === | ||
# Determine the maximum value of <math>m^2 + n^2 </math>, where <math>m </math> and <math>n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math>( n^2 - mn - m^2 )^2 = 1 </math>. <div style="text-align:right">([[1981 IMO Problems/Problem 3|1981 IMO, Problem 3]])</div> | # Determine the maximum value of <math>m^2 + n^2 </math>, where <math>m </math> and <math>n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math>( n^2 - mn - m^2 )^2 = 1 </math>. <div style="text-align:right">([[1981 IMO Problems/Problem 3|1981 IMO, Problem 3]])</div> |
Revision as of 16:23, 6 December 2007
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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. Often, there is a zero-th term added in equal to 0. The first few terms are
.
The Fibonacci sequence can be written recursively as . There is also an explicit definition below.
Contents
[hide]Phi
Ratios between successive terms, , , , , , tend towards the limit phi.
Binet's formula
Binet's formula is an explicit formula used to find the th term of the Fibonacci sequence. It is .
Proof: If we experiment with fairly large numbers, we see that the quotient of consecutive terms of the sequence approach (see above). Thus we have a sequence resembling that of a geometric sequence, which we let be . Then, . Using the quadratic formula, we find .
We now have two sequences and , but neither match up with the Fibonacci sequence. In particular, , but for to be zero, we need , but then the sequence just generates a constant . After a bit of experimenting with these two sequences to find a sequence where the zeroth term being zero, notice that , so also satisfies this recurrence. If we match up the numbers of and , we note that . However, , which implies that . Now, satisfies the same recurrence as and has the same initial terms, so we are done.
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?
- Find .
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)
- Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer produces a sequence of maximum length?
- A fair coin is to be tossed times. Let , in lowest terms, be the probability that heads never occur on consecutive tosses. Find .
- Find if and are integers such that is a factor of .
Olympiad
- Determine the maximum value of , where and are integers satisfying and .
See also
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