Sequences and Induction

1. Sequences and Induction Kavita Hatwal Fall 2002

2. Sequences Imagine that a person starts to count his ancestors.He has 2 parents, 4 grandparents, 8 grand grandparents and so forth. Writing this info in a row will look like 2, 4, 8, 16, 32, 64, … where … is called an ellipse and stands for so forth. So if generation is 1’st the grandparents are 2 if generation is 2’nd the grandparents are which is 4 if generation is 3’rd the grandparents are which is 8 . . . if generation is k’th the grandparents are Kavita Hatwal Fall 2002

3. So if be the number of grandparents for k’th generation, then We can find the value of for a particular value of k by applying the rule on the right hand side (RHS). A sequence informally is a set of elements written in a row. Each individual element (read a sub k) is called a term. The k is called a subscript or index. The m is the subscript of the initial term. The n is the subscript of the final term. Apply the above knowledge to the ancestor example. Denotes an infinite sequence Kavita Hatwal Fall 2002

4. Finding terms of sequence given by explicit formula Using the formulas for Page 192, # 2, 4 Page 213 #2, 5 (new book) Kavita Hatwal Fall 2002

5. Finding an explicit formula to fit given initial terms Page 192, # 13, 14 Page 213 #13, 15 (new book) Summation Notation Consider again To find the total # of ancestors for past 6 generations, we’ve to do What is the sum of the first 6 generations’ ancestors? In short hand the above sum is written as Summation symbol sigma starting or the lower limit of the sum ending or the upper limit of the sum More generally Page 192, # 18-a,c Page 213 # 18-a,c(new book) Kavita Hatwal Fall 2002

6. Summation given by formula • Page 192, # 21 • Page 213 # 21 (new book) • Changing from summation notation to expanded form • Page 192, # 27 • Page 213 #30 (new book) • Changing from expanded form to summation notation • Page 192, #31 • Page 213 #41(new book) • Change of variable and transformation of sum by change of variable • The index of summation is called a dummy variable. The appearance of summation can be altered by more complicated changes of variable as well. • Transformation of sum by change of variable. • change of variable: j = k+1 • Calculate the new lower and upper limits. • Calculate the general term of the new summation. • Put 1. and 2. Together • EXTRA TWIST. Replace the original variable wherever the new variable appears, leaving other things unchanged • change of variable: j = k-1 Kavita Hatwal Fall 2002

7. Product notation Just like summation notation there is product notation where we multiply instead of add the terms Page 192, #22 Page 213 #22 (new book) Read table 4.1.1 , page 187 on your own Factorial notation For each positive integer n, the quantity n factorial, denoted n! is defined as n! = n . (n-1) . (n-2)…3 . 2 . 1 0! = 1 n! = n . (n-1)! • Find • 3!, 5! • 8!/6! (n+1)!/(n-3)! Kavita Hatwal Fall 2002

8. INDUCTION • The principle of mathematical induction is used to check conjectures about the outcomes of processes that occur repeatedly and according to definite proportions. • Inductive proofs are based on the idea that you want to prove an infinite sequence of statements: • property p is true for number 1.property p is true for number 2.property p is true for number 3 • property p is true for number n • In other words, if you can prove p is true for 1 and • If p is true for n, then p is true for n + 1 for all natural numbers n. Kavita Hatwal Fall 2002

9. Assume you want to prove that for some statement P, P(n) is true for all n starting with n = 1. The Principle (or axiom) of Math Induction states that for this purpose you should accomplish just two steps: • Prove that P(1) is true. • Assume that P(k) is true for some k. Derive from here that P(k+1) is also true. • The idea here is that a finite number of steps may be needed to prove an infinite number of statements P(1), P(2), P(3), .... It is often compared to the domino effect. Imagine a collection of dominoes positioned one behind the other in such a way that if any given domino falls backward, it makes the one behind it fall backward also. • Let P(n) = ‘the n’th domino falls backward’ • If P(1) is true, that is the first domino falls backward • if P(k) , k >=1, that is the k’th domino falls backward is true • Then P(k+1) is true, that is the (k+1)’st domino falls backward Kavita Hatwal Fall 2002

10. Let P(n) = 1 + 3 + 5 + ... + (2n-1) = n2 • (1) Is P(1) true? • Assume that for an arbitrary k, P(k) is also true, i.e. 1 + 3 + ... + (2k-1) = k2. • Let's derive P(k+1) from this assumption • We have • 1+3+...+(2k-1)+(2k+1) • = [1+3+...+(2k-1)] + (2k+1) •  = k2 + (2k+1) •  = (k+1)2 • Let P(n) = 1 + 2 + 3 + …+ n = • Is P(1) true? If yes, then • Assume that for an arbitrary k, P(k) is also true, i.e. • Let's derive P(k+1) from this assumption • We have • P(k+1) = (1 + 2 + 3 + …+ k)+(k+1) • = k(k+1)/2 + (k+1) • =? • Page 204, #4 • Page 226 #4(new book) Kavita Hatwal Fall 2002

11. Arithmetic sequence A sequence in which each term is obtained by adding a fixed factor to the preceding term. Example if Geometric sequence A sequence in which each term is obtained by multiplying a fixed factor to the preceding term. Example if Sum of geometric sequence is given as Page 205, #20 Page 226 #20(new book) Kavita Hatwal Fall 2002