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### High School Mathematics - 23.9 Fibonacci Sequence and Finite and Infinite Sequences

 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: a0 = 1, a1 = 1, ak=ak-2+ak-1 The first six terms of this sequence are: a0 = 1 a1 = 1 a2 = a0 + a1 = 1 + 1 = 2 a3 = a1 + a2 = 1 + 2 = 3 a4 = a2 + a3 = 2 + 3 = 5 a5 = a3 + a4 = 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 a1 = 1000, and d = -15 We can calculate a30 = 1000 - 15(30 - 1) = 565 Now we can use the formula S30 = 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 a1 = 40,000, a2 = 40,000(1.03) = 41,200, a3 = 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.
 Q 1: __________ is the sequence where the starting numbers (or seeds) are 1 and 1, and we add the two previous numbers to get the next number in the sequence.Finite SequenceFibonacci sequenceInfinite Sequence Q 2: Choose the infinite series from the following: 4 + 7 + 10 + 13 + 162 + 4 + 6 + 8 + 10 .... +247 + 10 + 13 + 16 + 19 + 21 + 24 + ... Q 3: 5, 10, 15, 20, . . . . . 100 is an example of finite seriesinfinite seriesfinite sequenceinfinite sequence Q 4: A ___________ sequence is a sequence whose domain consists of the set {1, 2, 3, ... n} or in other words the first n positive integers. finiteinfinite Question 5: This question is available to subscribers only! Question 6: This question is available to subscribers only!