Recursive Sequences Terry Anderson

What is a Recursive Sequence? A sequence that follows a pattern involving previous terms  To generate new terms, we use old ones  The terms t 1 (and sometimes t 2, t 3 …) must be defined ahead of time

Example of a Recursive Sequence Define t 1 = 3 General term: t n = 2t n  Note that t n-1 is the term before t n Now we may find the other terms  t 2 = 2t = 2(3) + 1 = 7  t 3 = 2t = 2(7) + 1 = 15  t 4 = 2t = 2(15) + 1 = 31

Fibonacci’s Rabbits 1202: Italian mathematician Leonardo Fibonacci publishes his solution to the famous problem:  “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” Or, in modern English:  We start with one pair of rabbits  A pair of rabbits take one month to mature  Each pair of mature rabbits mate to produce another pair of rabbits every month

Month (n) Number of Pairs (t n )

The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …  Each term is the sum of the previous two  Define t 1 = 1, t 2 = 1  General term: t n = t n-1 + t n-2 This sequence appears in many other places in nature…

The most common number of petals on flowers belong to the sequence… 1 petal (white calla lily) 2 petals (euphorbia) 3 petals (trillium) 5 petals (columbine) 8 petals (bloodroot) 13 petals (black-eyed susan) 21 petals (shasta daisy) 34 petals (field daisy)

The patterns found on pineapples, pinecones, sunflowers, etc. can also be seen in the sequence…

The Golden Ratio We can find this number by dividing successive terms of the Fibonacci Sequence:  2/1=1  3/2=1.5  5/3=1.667  8/5=1.6  13/8=1.625  21/13=1.615  34/21=1.619  And so on… At infinity, this ratio converges to the golden ratio,  (phi):  appears in…  Nature  Paintings  Classical architecture  The pyramids  The human body  Music  And many other places…