Download 01.03. Fibonacci Sequence

Fibonacci Sequence
Fibonacci’s problem
• About 800 years ago he wrote a book about
algebra with this problem…
Text p. 9 Procedure A to D
Remember:
• Pairs of rabbits (not individuals)
• Adult rabbits can reproduce during second
month and after, baby rabbits have to wait
• AA = pair of adult rabbits
• BB = pair of baby rabbits
Procedure A
Start of
month 1
BB
Start of
month 2
AA
1 pair
Start of
month 6
BB, AA,
AA, BB,
AA, AA,
BB, AA
1 pair
2 pairs
Start of month 7
AA, BB, AA, BB, AA,
AA, BB, AA, AA, BB,
AA, AA, BB
8 pairs
13 pairs
Start of
month 3
AA, BB
Start of
Start of
month 4
month 5
BB, AA, AA AA, BB,
AA, AA, BB
3 pairs
5 pairs
Start of month 8
BB, AA, AA, BB, AA, AA,
BB, AA, AA, BB, AA, BB,
AA, AA, BB, AA, BB, AA
AA, BB, AA
21 pairs
Procedure B
Sequence: {1, 1, 2, 3, 5, 8, 13, 21, …}
Procedure C
Pattern: term is the sum of the two previous terms
13 terms are: {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233}
Procedure D
13 terms represent one year of reproduction,
therefore there will be 233 pairs of rabbits when
the original pair turns one year old, at the start of
month 13
Fibonacci & Nature
• Video
Practice: Text p. 9 Q 19-22
Q19a) Third number is the sum of the previous
two
b) Ex. remove the first 10 terms
The remaining number form a Fibonacci-like
sequence where each term is the sum of the
two previous
{89, 144, 233, 377, 610, 987, 1597, 2584,
4181, 6765, …}
Q20a) 19 and 20 are consecutive…
4181 + 6765 = 10 946, (21st Fibonacci #)
b) The difference between 20th and 19th!
6765 - 4181 = 2584, (18th Fibonacci #)
Q21) Recursive pattern, so find the 50th by
continuing to find the sum of consecutive
terms
50th Fibonacci # = 12586269025
Q22a) each term is the sum of the previous two
terms; each term is a Fibonacci number.
Petal
count
b)
2
3
5
8
13
21
34
55
89
Plant
Check your understanding
Text p. 10 Q 23-29
• Q23. use a LIST editor on graphing calculator
Q23a) {2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288,
466, 754, 1220}
b) {6, 6, 7, 8, 10, 13, 18, 26, 39, 60, 94, 149, 238,
382, 615}
c) {0.5, 0.5, 1, 1.5, 2.5, 4, 6.5, 10.5, 17, 27.5, 44.5,
72, 116.5, 188.5, 305}
d) Multiplying & dividing by constants (A & C) act
like Fibonacci.
Adding and subtracting by a constant does not.
Q24a) The greater the pair of Fibonacci
numbers, the closer the quotient gets to 0.618
Fibo 1 1
nacc
i#
2
Quo 1 0.5 0.666
tien
667
t
3
5
8
13
21
34
55
89
0.6 0.625 0.615 0.619 0.61 0.61 0.61 0.61
385 048
7
8
7
8
647 182 978 056
Q25a) Any rectangle with something close to
ratio
Ex. 10cm by 16cm
b) Answers vary, close to Golden Ratio = 0.618
c) Picture frames, labels, ads, index cards,
playing cards, light switch plates, credit cards,
bank cards, etc.
Q26c) The greater the pair of Fibonacci
numbers, the closer the quotient gets to 1.618
Fibo 1 1
nacc
i#
2
3
5
8
13
21
34
55
89
Quo 1 2
tien
t
1.666
667
1.6 1.625 1.615 1.619 1.61 1.61 1.61 1.61
385 048
7
8
7
8
647 182 978 056
Q27) Numbers in measurements are Fibonacci
numbers
Line segment AB
Length (mm) 34
FA
55
FJ
89
FI
144
Q28a) 2, 4, 7, 12, 20, 33
b) {2, 4, 7, 12, 20, 33, …}
{4-2, 7-4, 12-7, 20-12, 33-20}
{2, 3, 5, 8, 13, …}
c) All differences are Fibonacci #’s
Q29a) 3 x 5
5x8
8 x 13
b) The sum of the squares of the first n
consecutive Fibonacci #’s is the product of the
nth and (n+1)th Fibonacci number
• Donald Duck and the Golden Ratio
• Phi’s the Limit