Sequences and Summations ICS 6D. Prof. Sandy Irani

Transcription

1 Sequences and Summations ICS 6D Prof. Sandy Irani

2 Test Poll What is your favorite flavor of ice cream? A) Vanilla B) Chocolate C) Moose Tracks D) Mint Chocolate Chip E) None of the above

3 Sequences A sequence is a special case of a function in which the domain is a consecutive set of integers: For example: a person s height measured in inches on each birthday. (The person for this example if 5 years old). h 0 = 20, h 1 = 30, h 2 = 34, h 3 = 37, h 4 = 40, h 5 = 43 What is the fourth term of the sequence? What is the word for the 3 in h 3? What is the notation for the entire sequence.

4 Sequences A finite sequence has a finite domain: h 0 = 20, h 1 = 30, h 2 = 34, h 3 = 37, h 4 = 40, h 5 = 43 An infinite sequence has an infinite domain: h 2 = 1, h 3 = 2, h 4 = 4, h 5 = 8, h 6 = 16, h 7 = 32,..

5 Ways to specify a sequences List the numbers. h 0 = 20, h 1 = 30, h 2 = 34, h 3 = 37, h 4 = 40, h 5 = 43 Average rainfall in California since Average weight of a child in the US by age. Give an explicit formula for the sequence. Example: Initial index = 1. f n = 2 n + 2n f 1 = 4, f 2 = 8, f 3 = 14, f 4 = 24, f 5 = 42,..

6 Ways to specify a sequences Recurrence relation Base case: one or more initial values f 0 = 1 f 1 = 1 Recursive rule: a rule for determining the next term in the sequence as a function of terms that appear earlier in the sequence (recurrence relation): f n = f n 1 +f n 2,for n 2

7 Fibonacci Sequence f 0 = 1 f 1 = 1 f n = f n 1 +f n 2,for n 2

8 More recurrence relations: g 0 = 0 g n = g (n div 2) + 1, for n 1. n div 2 = n/2

9 More recurrence relations: h 0 = 2 h 1 = 4 h n = n h n 1 + 2h n 2, for n 2 Need enough initial values in the base case to specify the sequence

10 Recurrence Relations Useful for modeling dynamical systems: Biology: population growth Finance: market growth or interest accumulation Useful in computer science The number of operations performed by a recursive algorithm on an input of a given size.

11 Fibonacci Sequence f 0 = 1 f 1 = 1 f n = f n 1 +f n 2,for n 2 Developed by Leonardo Fibonacci to describe the population growth of a colony of rabbits. How big is the rabbit colony after 100 months? How long does it take the rabbit colony to reach 1000 rabbits?»need an explicit formula for f n

12 Solving a Recurrence Given a recurrence relation that defines a sequence Initial values Recursive Rule Find an explicit formula for the sequence: f n = a mathematical function that depends only on n and not on earlier terms in the sequence

13 Geometric Sequence g 0 = a (a = initial value) g n = r g n 1 (r = common ratio), for n 1.

14 Geometric Sequence Examples a = 5, r = 2 a = 1, r =.1 a = 2, r = 1 g 0 = 4, g 1 = 12, g 2 = 36, g 3 = 108,.

15 Geometric Sequence g 0 = a (a = initial value) g n = r g n 1 (r = common ratio), for n 1. What are the conditions that make {g n } increasing? What are the conditions that make {g n } decreasing?

16 Arithmetic Sequence h 0 = a (a = initial value) h n = d + h n 1 (d = common difference) for n 1.

17 Arithmetic Sequence Examples a = 5, d = 2 a = 1, d =.1 a = 2, d = 1 g 0 = 4, g 1 = 9, g 2 = 14, g 3 = 19,.

18 Arithmetic Sequence h 0 = a (a = initial value) h n = d + h n 1 (d = common difference) for n 1. What are the conditions that make {h n } increasing? What are the conditions that make {h n } decreasing?

19 Summations Compact expression for the sum of the terms in a mathematical sequence. + +

20 Summations If the terms in the sequence have an explicit formula, then the value of the sum can be computed: Parentheses are important:

21 Summations If the upper limit is a variable, then the value of the sum is a function of that variable:

22 Summations Sometimes, we need to pull out one or more terms from the summation.

23 Summations A closed form for a summation, expresses the value of the sum (without a summation)

24 Closed form for the sum of terms in a geometric sequence Geometric sequence: Initial value a, common ratio r, n = # of terms: What is the value of

25 Closed form for the sum of terms in an arithmetic sequence Arithmetic sequence: Initial value a, common difference d, n = # of terms: What is the value of?