1 CMSC 250 Chapter 4, Summations and Products

2 CMSC 250 Induction l Induction is a proof technique used to verify a property of a sequence –2,4,6,8,… for i  1 a i = 2i an infinite sequence with infinite distinct values –for i  1 b i = (  1) i an infinite sequence with finite distinct values –for 1  i  6 c i = i + 5 a finite sequence (with finite distinct values)

3 CMSC 250 Finding an explicit formula l Figure out the formula for this sequence:

4 CMSC 250 Finding an explicit formula l Different sequences with the same initial values:

5 CMSC 250 Summation & product notation l Sum of items specified l Product of items specified

6 CMSC 250 Variable ending point l n as the index of the final term l for n = 2 l for n = 3

7 CMSC 250 Nesting of sum/product notation l Variations (same or different??):

8 CMSC 250 Telescoping series

9 CMSC 250 Properties l Merging and splitting

10 CMSC 250 Properties, con't. l Distribution

11 CMSC 250 Discrete Structures CMSC 250 Lecture 22 March 24, 2008

12 CMSC 250 Factorial l n! = n  (n  1)  (n  2)  …  2  1 l Definition: 0! = 1 n! = n  (n  1)!