Download presentation

1
**Sequences and Mathematical Induction**

Chapter 4 Sequences and Mathematical Induction

2
4.1 Sequences

3
Sequences The main mathematical structure used to study repeated processes is the sequence. The main mathematical tool used to verify conjectures about patterns governing the arrangement of terms in sequences is mathematical induction.

4
**Example Ancestor counting with a sequence**

two parents, four grandparents, eight great-grandparents, etc. Number of ancestors can be represented as 2position Example: 23 = 8 (great grandparents), therefore parents removed three generations are great grandparents for which you have a total of 8.

5
Sequences Sequence is a set of elements written in a row as illustrated on prior slide. (NOTE: a sequence can be written differently) Each element of the sequence is a term. Example am, am+1, am+2, am+3, …, an terms a sub m, a sub m+1, a sub m+2, etc. m is subscript of initial term n is subscript of final term

6
**Example Finding terms of a sequence given explicit formulas**

ak = k/(k+1) for all integers k ≥ 1 bi = (i-1)/i for all integers i ≥ 2a the sequences a and b have the same terms and hence, are identical a1 = 1/(1+1) = ½ b2 = (2-1)/2 = ½ a2 = 2/(2+1) = 2/3 b3 = (3-1)/3 = 2/3 a3 = 3/(3+1) = 3/4 b4 = (4-1)/4 = 3/4 a4 = 4/5 b5 = 4/5

7
**Example Alternating Sequence cj = (-1)j for all integers j≥0**

sequence has bound values for the term. term ∈ {-1, 1} c0 = (-1)0 = 1 c1 = (-1)1 = -1 c2 = (-1)2 = 1 c3 = (-1)3 = -1 c4 = (-1)4 = 1 …

8
**Example Find an explicit formula to fit given initial terms**

sequence = 1, -1/4, 1/9, -1/16, 1/25, -1/36, … What can we observe about this sequence? alternate in sign numerator is always 1 denominator is a square ak = ±1 / k2 (from the previous example we know how to create oscillating sign sequence, odd negative and even positive. ak = (-1)k+1 / k2 1/12 -1/22 1/32 -1/42 1/52 -1/62 a1 a2 a3 a4 a5 a6

9
Summation Notation Summation notation is used to create a compact form for summation sequences governed by a formula. the sequence is governed by k which has lower limit (1) and a upper limit of n. This sequence is finite because it is bounded on the lower and upper limits.

10
**Example Computing summations**

a1 = -2, a2 = -1, a3 = 0, a4 = 1, and a5 =2.

11
Example Computing summation from sum form.

12
Example Changing from Summation Notation to Expanded form

13
**Example Changing from expanded to summation form.**

Find a close form for the following:

14
**Separating Off a Final Term**

A final term can be removed from the summation form as follows. Example of use: Rewrite the following separating the final term

15
Example Combining final term

16
**Telescoping Sum Telescoping sum can be evaluated to a closed form.**

See book page 205

17
Product Notation Recursive form

18
Example Compute the following products:

19
Factorial Factorial is for each positive integer n, the quantity n factorial denoted n! is defined to be the product of all the integers from 1 to n: n! = n * (n-1) *…*3*2*1 Zero factorial denoted 0! is equal to 1.

20
Example Computing Factorials

21
**Properties of Summations and Products**

Theorem 4.1.1 If am, am+1, am+2, … and bm, bm+1, bm+2, … are sequences of real numbers and c is any real number, then the following equations hold for any integer n≥m:

22
Examples Let ak = k +1 and bk = k – 1 for all integers k

23
Examples Let ak = k +1 and bk = k – 1 for all integers k

24
**Transforming a Sum by Change of Variable**

Transform the following by changing the variable. summation: change of variable: j = k+1 Solution: compute the new limits: lower: j=k+1, j=0+1=1 upper: j=k+1, j=6+1=7

Similar presentations

OK

Slide 11.1- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Slide 11.1- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on ideal gas law definition Ppt on central administrative tribunal bench Ppt on autonomous car sensors Ppt on media literacy Ppt on soft contact lenses Ppt on nature and scope of production and operation management Ppt on ready to serve beverages and more locations Ppt on obesity management association Ppt on conceptual artwork Ppt on cnc lathe machine