1. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity 11 Basic Structure : Sets, Functions, Sequences, and Sums Sequences and Summations

2. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Sequences Sequences represent ordered lists of elements. A sequence is defined as a function from a subset of N ( either {0,1,2,…} or {1,2,3,…} to a set S. We use the notation a n to denote the image of the integer n. We call a n a term of the sequence. Example: subset of N: 1 2 3 4 5 … S: 2 4 6 8 10 …

3. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Sequences We use the notation {a n } to describe a sequence. Important: Do not confuse this with the {} used in set notation. It is convenient to describe a sequence with a formula. For example, the sequence on the previous slide can be specified as {a n }, where a n = 2n.

4. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Sequences Example: Consider the sequence {a n }, where a n =1/n The list of the term of this sequence, beginning with a 1, namely a 1,a 2,… Starts with 1,1/2,1/3,…

5. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity The Formula Game 1, 3, 5, 7, 9, … a n = 2n - 1 3, 5, 9, 17, 33, … a n = 2 n + 1 0.25, 0.5, 0.75, 1, 1.25 … a n = 0.25n 3, 9, 27, 81, … a n = 3 n What are the formulas that describe the following sequences a 1, a 2, a 3, … ?

6. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Sequences A geometric progression is a sequence of the form a, ar, ar 2, ar 3, …, ar n, …. Where the initial term a and the common ration r are real numbers For example : The sequence WithInitial termCommon term Start n=0 a0,a1,a2,… {b n }b n =(-1) n 11,-1,1,-1,1,…. {c n }c n =2*5 n 252,10,50,250,…. {d n }d n =6*(1/3) n 61/36,2,2/3,2/9,….

7. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Sequences An arithmetic progression is a sequence of the form a, a+d, a+2d, a+3d, …, a+nd, …. Where the initial term a and the common difference d are real numbers For example : The sequence WithInitial termCommon term Start n=0 a0,a1,a2,… {s n }S n =-1+4n4-1,3,7,11,… {t n }t n =7-3n7-37,4,1,-2,…

8. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity The Formula Game 1, 1/2, 1/4, 1/8, 1/16,… a n = 1/2 n n=0,1,2,… geometric progression a= 1 r=1/2 r=1/2 1,3,5,7,…. an = 2n+1 n=0,1,2,… n=0,1,2,… arithmetic progression a= 1 d=2 d=2 What are the formulas, and find what kind of progression ?

9. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Some useful Sequences Nth termFirst 10 Terms a 1,a 2,a 3 n2n2 1,4,9,16,25,36,49,64,81,…. n3n3 1,8,27,64,125,216,…. n4n4 1,16,81,256,625,… 2n2n 2,4,8,16,32,64,128,… 3n3n 3,9,27,81,243,729,… n!1,2,6,24,120,720,…

10. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Strings Finite sequences are also called strings, denoted by a 1 a 2 a 3 …a n. The length of a string S is the number of terms that it consists of. The empty string contains no terms at all. It has length zero.

11. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Summations It represents the sum a m + a m+1 + a m+2 + … + a n. The variable j is called the index of summation, running from its lower limit m to its upper limit n. We could as well have used any other letter to denote this index. What does stand for?

12. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Summations It is 1 + 2 + 3 + 4 + 5 + 6 = 21.. We write it as. What is the value of ? How can we express the sum of the first 1000 terms of the sequence {a n } with a n =n 2 for n = 1, 2, 3, … ? It is 1 + 4 + 9 + 16 + 25 = 55.

13. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Summations  Sometimes it is useful to shift the index of summation in a sum, suppose:  We want the index to run from 0 to 4 rather than 1 to 5  Let k = j – 1, then j 2 = (k+1) 2

14. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Useful Summations formula The following formula: When you have such a formula, the result of any summation can be calculated much more easily, for example:

15. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Geometric Series If a and r are real numbers and r≠0, then (n+1)a If r≠1 If r=1 Proof: Let S = and continue…

16. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Useful Series 1. 2. 3.

17. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Double Summations Corresponding to nested loops in C or Java, there is also double (or triple etc.) summation: Example:

18. Fall 2008/2009 I. Arwa Linjawi & I. Asma’a Ashenkity Summations Example :Find += = - = n(n+1)(2n+1)/6 from formula 2 = 100*101*201 – 49*50*99 = 338,350 – 40,425 6 6