The Fibonacci sequence was discovered by Leonardo of Pisa. This sequence occurs in nature, and Leonardo of Pisa derived it by studying the mating patterns of rabbits.
The Fibonacci sequence is defined recursively, as follows:
a_{0} = 1, a_{1} = 1, a_{k}=a_{k2}+a_{k1}
The first six terms of this sequence are:
a_{0} = 1
a_{1} = 1
a_{2} = a_{0} + a_{1} = 1 + 1 = 2
a_{3} = a_{1} + a_{2} = 1 + 2 = 3
a_{4} = a_{2} + a_{3} = 2 + 3 = 5
a_{5} = a_{3} + a_{4} = 3 + 5 = 8
Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ........
Finite and Infinite Sequences
A sequence is called finite if the number of terms is finite. A finite sequence has always a last term.
Examples:
2, 5, 8, 11, 14 …, 32
37, 33 …, 1
A sequence is called infinite if the number of terms is infinite. An infinite sequence has no last term. In this sequence, every term is followed by a new term.
Examples:
i) A sequence of multiples of 5
5, 10, 15, 20, …
ii) A sequence of reciprocals of positive integers
1, 1/2, 1/3, 1/4, 1/5 . . . . .
The above two sequences are clearly the infinite sequences.
 Find a formula for the nth term of the arithmetic sequence: 2, 8, 14, 20, .....
 Calculate the missing terms in the arithmetic sequence, if a1 = 6, and a5 = 66.
 Application
Suppose that Harry play black jack at Harrah's on June 1 and lose $1,000. Tomorrow Harry bets and lose $15 less. Each day he loses $15 less that his previous loss. What will be his total losses for the 30 days of June?
Solution
This is an arithmetic series with a_{1} = 1000, and d = 15
We can calculate a_{30} = 1000  15(30  1) = 565
Now we can use the formula S_{30} = 30/2 (1000 + 565) = 23,475
Harry will lose a total of $23,475 during June.
 Lisa's starting salary was $40,000 per year. Each year she receive a cost of living adjustment (COLA) of three percent of our original salary. Write a sequence showing her salary for her first five years of working in the company.
Solution
We have a_{1} = 40,000, a_{2} = 40,000(1.03) = 41,200, a_{3} = 41,000(1.03) = 40,000(1.03)2 = 42,436, a4 = 40,000(1.03)3 = 43,709.08, a5 = 40,000(1.03)4 = 45,020.35
Directions: Answer the following questions. Also write at least 5 examples of your own.
