Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1."

Transcription

1 Chapter 4. Suppose you wish to prove that the followig is true for all positive itegers by usig the Priciple of Mathematical Iductio: ( ) =. (a) Write P() (b) Write P(7) (c) Write P(73) (d) Use P(7) to prove P(73) (e) Write P(k) (f) Write P(k + ) (g) Use the Priciple of Mathematical Iductio to prove that P() is true for all positive itegers As: (a) =. (b) = 7. (c) = 73. (d) = ( ) + 45 = = = (7 + ) = 73. (e) (k ) = k. (f) (k + ) = (k + ). (g) P() is true sice =. P(k) P(k + ): (k + ) = k + (k + ) = (k + ).. Suppose you wish to use the Priciple of Mathematical Iductio to prove that! +! + 3 3! ! = ( + )! for all. (a) Write P() (b) Write P(5) (c) Write P(k) (d) Write P(k + ) (e) Use the Priciple of Mathematical Iductio to prove that P() is true for all As: (a)! =!. (b)! +! ! = 6!. (c)! +! k k! = (k + )!. (d)! +! (k + )(k + )! = (k + )!. (e) P() is true sice! = ad! =. P(k) P(k + ):!+!+...+ ( k+ )( k+ )! = ( k+ )! + ( k+ )( k+ )! = ( k+ )[! + ( k+ )] = ( k+ )(! k+ ) = ( k + )!. Page 55

2 3. Use the Priciple of Mathematical Iductio to prove that + 3 ( ) ( ) = for all positive itegers. 3 ( ) + As: P(): =, which is true sice both sides are equal to. P(k) P(k 3 + ): k+ k k+ k k+ k+ k+ k+ ( ) + k+ k+ ( ) + + 3( ) ( ) = + ( ) = 3 3 k + k k ( ) ( 3( )) + k k ( ) ( ) + k ( ) + = = = Use the Priciple of Mathematical Iductio to prove that + 3 for all. As: P(): + 3, which is true sice both sides are equal to 3. P(k) P(k + ): + k + = ( + k ) + k 3 k + k 3 k + 3 k = 3 k < 3 3 k = 3 k Use the Priciple of Mathematical Iductio to prove that 3 > + 3 for all. As: P(): 3 > + 3 is true sice 8 > 7. P(k) P(k + ): (k + ) + 3 = k + k = (k + 3) + k + < k 3 + k + k 3 + 3k k 3 + 3k + 3k + = (k + ) Use the Priciple of Mathematical Iductio to prove that ( + ) for all 0. As: P(0): 0 + 0, which is true sice 0. P(k) P(k + ): (k + ) + (k + ) = (k + k) + (k + ), which is divisible by sice k + k ad (k + ) Use the Priciple of Mathematical Iductio to prove that = for all 0. 3 As: P(0): =, which is true sice =. P(k) P(k + ): k+ k+ k+ k+ 3 k k = + 3 = =. Page 56

3 8. Use the Priciple of Mathematical Iductio to prove that (3 ) (3 ) = for all. As: P(): =, which is true sice =. P(k) P(k + ): k(3k ) (3( k+ ) ) = + (3k+ ) k(3k ) + (3k+ ) 3k + 5k+ = = (3k+ )( k+ ) ( k+ )(3( k+ ) ) = = 9. Use the Priciple of Mathematical Iductio to prove that ( + 3) for all. As: P(): + 3, which is true sice 4. P(k) P(k + ): (k + ) + 3(k + ) = (k + 3k) + (k + ), which is divisible by sice k + 3k ad (k + ). 0. Use the Priciple of Mathematical Iductio to prove that + 3 for all 4. As: P(4): , which is true sice 6. P(k) P(k + ): (k + ) + 3 = (k + 3) + k + k + k = k +.. Use the Priciple of Mathematical Iductio to prove that 3 ( ) for all. As: P(): , which is true sice 3 6. P(k) P(k + ): (k + ) 3 + 3(k + ) + (k + ) = (k 3 + 3k + k) + 3(k + 3k + ), which is divisible by 3 sice each of the two terms is divisible by 3.. Use the Priciple of Mathematical Iductio to prove that ay iteger amout of postage from 8 cets o up ca be made from a ifiite supply of 4-cet ad 7-cet stamps. As: P(8): use oe 4-cet stamp ad two 7-cet stamps. P(k) P(k + ): if a pile of stamps for k cets postage has a 7-cet stamp, replace oe 7-cet stamp with two 4- cet stamps; if the pile cotais oly 4-cet stamps (there must be at least five of them), replace five 4-cet stamps with three 7-cet stamps. 3. Suppose that the oly currecy were 3-dollar bills ad 0-dollar bills. Show that ay dollar amout greater tha 7 dollars could be made from a combiatio of these bills. As: P(8): Eightee dollars ca be made usig six 3-dollar bills. P(k) P(k + ): Suppose that k dollars ca be formed, for some k 8. If at least two 0-dollar bills are used, replace them by seve 3-dollar bills to form k + dollars. Otherwise (that is, at most oe 0-dollar bill is used), at least three 3-dollar bills are beig used, ad three of them ca be replaced by oe 0-dollar bill to form k + dollars. Page 57

4 4. Use mathematical iductio to prove that every amout of postage of six cets or more ca be formed usig 3-cet ad 4-cet stamps. As: P(6): Six cets postage ca be made from two 3-cet stamps. P(k) P(k + ): either replace a 3-cet stamp by a 4-cet stamp or else (if there are oly 4-cet stamps i the pile of stamps makig k cets postage) replace two 4-cet stamps by three 3-cet stamps. 5. Prove that ( j+ ) = 3 for all positive itegers. j= As: The basis case holds sice k j= k ( k + ) j= k+ ( j+ ) = 3k k ( j + ) = 3 = 3. Now assume that j= for some k. It follows that ( j+ ) = ( j+ ) (k+ ) + (4k+ ) + (4k+ 3) = 3k + 6k+ 3 = 3( k+ ). j= k 6. Use mathematical iductio to show that lies i the plae passig through the same poit divide the plae ito regios. As: The basis step follows sice oe lie divides the plae ito regios. Now assume that k lies passig through the same poit divide the plae ito k regios. Addig the (k + )st lie splits exactly two of these regios ito two parts each. Hece k + cocurret lies split the plae ito k + = (k + ) regios. 7. Let a =, a = 9, ad a = a + 3a for 3. Show that a 3 for all positive itegers. As: Let P() be the propositio that a 3. The proof uses the Secod Priciple of Mathematical Iductio. The basis step follows sice a = 3 = 3 ad a = 9 9 = 3. Now assume that P(k) is true for all k such that k <. The a k 3 k for k <. Hece a = a + 3a = = 3 3 = Floor borders oe foot wide ad of varyig legths are to be covered with ooverlappig tiles that are available i two sizes: 3 ad 5 sizes. Assumig that the supply of each size is ifiite, prove that every border ( > 7) ca be covered with these tiles. As: P(8): use oe of each type. P(k) P(k + ): If a 5 tile is used as part of the coverig of a k strip, replace a 5 tile with two 3 tiles to cover a ( k + ) strip. Otherwise, the tiles for the k strip must iclude three 3 tiles; replace three of these with two 5 tiles to cover a ( k + ) strip. Page 58

5 9. A T-omio is the tile that is pictured below. Prove that every ( > ) chessboard ca be tiled with T-omioes. As: P(): The figure below shows a tilig of a 4 4 board. P(k) P(k + ): Divide the k + k + board ito four quarters, each of which is a k k board. P(k) guaratees that each of these four k k boards ca be tiled. Put these four tiled boards together to obtai a tilig for the k + k + board. 0. Use the Priciple of Mathematical Iductio to prove that 4 (9 5 ) for all 0. As: P(0): 4 is true sice 4 0. P(k) P(k + ): 9 k + 5 k + = 9(9 k 5 k ) + 5 k (9 5). Each term is divisible by 4: 4 9 k 5 k (by P(k)) ad Use the Priciple of Mathematical Iductio to prove that 5 (7 ) for all 0. As: P(): 5 7 is true sice 5 5. P(k) P(k + ): 7 k + k + = 7(7 k k ) + k (7 ). Each term is divisible by 5: 5 7 k k (by P(k)) ad Prove that the distributive law A (A... A ) = (A A )... (A A ) is true for all 3. As: The secod form of mathematical iductio is used. P(3) is true sice it is the ordiary distributive law for itersectio over uio. P(3)... P() P( + ): A (A... A + ) = A ((A... A ) A + ) = [A (A... A )] (A A + ) = [(A A )... (A A )] (A A + ) = (A A )... (A A + ) Prove that = for all. 4 8 ( ) As: P(): =, which is true sice the right side is equal to /. P(k) P(k + ): k+ k+ k+ k+ 3 k+ k k+ 4 k+ k+ 3 k ( k+ ) = + = = = k+ k k+ k+ k+ k Page 59

6 4. Fid the error i the followig proof of this theorem : Theorem: Every positive iteger equals the ext largest positive iteger. Proof: Let P() be the propositio ' = + '. To show that P(k) P(k + ), assume that P(k) is true for some k, so that k = k +. Add to both sides of this equatio to obtai k + = k +, which is P(k + ). Therefore P(k) P(k + ) is true. Hece P() is true for all positive itegers. As: No basis case has bee show. Use the followig to aswer questios 5-33: I the questios below give a recursive defiitio with iitial coditio(s). 5. The fuctio f () =, =,, 3,... As: f () = f ( - ), f () =. 6. The fuctio f () =!, = 0,,,... As: f () = f ( - ), f (0) =. 7. The fuctio f () = 5 +, =,, 3,... As: f () = f ( - ) + 5, f () = The sequece a = 6, a = 3, a 3 = 0, a 4 = 7,. As: a = a 3, a = The Fiboacci umbers,,, 3, 5, 8, 3,. As: a = a + a, a =, a =. 30. The set {0,3,6,9, }. As: 0 S; x S x + 3 S. 3. The set {,5,9,3,7, }. As: S; x S x + 4 S. 3. The set {,/3,/9,/7, }. As: S; x S x/3 S. 33. The set {, 4,,0,,4,6, }. As: 0 S; x S x ± S. Use the followig to aswer questios 34-39: I the questios below give a recursive defiitio (with iitial coditio(s)) of {a } ( =,,3, ). Page 60

7 34. a =. As: a = a, a =. 35. a = 3 5. As: a = a + 3, a =. 36. a = ( + )/3. As: a = a + /3, a = / a =. As: a = a, a =. 38. a = /. As: a =, a =. a 39. a = +. As: a = a +, a =. Use the followig to aswer questios 40-44: I the questios below give a recursive defiitio with iitial coditio(s) of the set S. 40. {3,7,,5,9,3, }. As: 3 S; x S x + 4 S. 4. All positive iteger multiples of 5. As: 5 S; x S x + 5 S. 4. {, 5, 3,,,3,5, }. As: S; x S x ± S. 43. {0.,0.0,0.00,0.000}. As: 0. S; x S x/0 S. 44. The set of strigs,,,,. As: S; x S x S (or x S 00x + S). 45. Fid f () ad f (3) if f () = f ( ) + 6, f (0) = 3. As: f () = 30, f (3) = Fid f () ad f (3) if f () = f ( ) f ( ) +, f (0) =, f () = 4. As: f () = 5, f (3) =. Page 6

8 47. Fid f () ad f (3) if f () = f ( ) / f ( ), f (0) =, f () = 5. As: f () = 5/, f (3) = /. 48. Suppose {a } is defied recursively by As: a 3 = 63 ad a 4 = 3,968. a = ad that a 0 =. Fid a 3 ad a 4. a 49. Give a recursive algorithm for computig a, where is a positive iteger ad a is a real umber. As: The followig procedure computes a: procedure mult(a: real umber, : positive iteger) if = the mult(a,) : = a else mult(a,) : = a + mult(a, ). 50. Describe a recursive algorithm for computig 3 where is a oegative iteger. As: The followig procedure computes 3 : procedure power(: oegative iteger) if = 0 the power() : = 3 else power() : = power( ) power( ). 5. Verify that the program segmet a : = b : = a + c is correct with respect to the iitial assertio c = 3 ad the fial assertio b = 5. As: Suppose c = 3. The program segmet assigs to a ad the assigs a + c = + 3 = 5 to b. 5. Cosider the followig program segmet: i : = total : = while i < begi i : = i + total : = total + i ed. ii ( + ) Let p be the propositio total = ad i. Use mathematical iductio to prove that p is a loop ivariat. As: Before the loop is etered p is true sice total = ad i. Suppose p is true ad i < after a executio of the loop. Suppose that the while loop is executed agai. The variable i is icremeted by, ad hece i. The variable total was ( i ) i, which ow becomes ( i ) i i( i+ ) i + =. Hece p is a loop ivariat. Page 6

9 53. Verify that the followig program segmet: if x y the max : = y else max : = x. is correct with respect to the iitial assertio T ad the fial assertio (x y max = y) (x > y max = x). As: If x < y iitially, max is set equal to y, so (x y max = y) is true. If x = y iitially, max is set equal to y, so (x y max = y) is agai true. If x > y, max is set equal to x, so (x > y max = y) is true. Page 63

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1 1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative

More information

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

1.3 Binomial Coefficients

1.3 Binomial Coefficients 18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

INDR 262 Optimization Models and Mathematical Programming LINEAR PROGRAMMING MODELS

INDR 262 Optimization Models and Mathematical Programming LINEAR PROGRAMMING MODELS LINEAR PROGRAMMING MODELS Commo termiology for liear programmig: - liear programmig models ivolve. resources deoted by i, there are m resources. activities deoted by, there are acitivities. performace

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr. Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

Section 1.6: Proof by Mathematical Induction

Section 1.6: Proof by Mathematical Induction Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Span, linear independence, bases, and dimension Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

More information

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem

Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem 116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

Sequences and Series Using the TI-89 Calculator

Sequences and Series Using the TI-89 Calculator RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator Norecursively Defied Sequeces A orecursively defied sequece is oe i which the formula for the terms of the sequece is give explicitly. For

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

Solving Divide-and-Conquer Recurrences

Solving Divide-and-Conquer Recurrences Solvig Divide-ad-Coquer Recurreces Victor Adamchik A divide-ad-coquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios

More information

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 )

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 ) Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

Solutions to Exercises Chapter 4: Recurrence relations and generating functions

Solutions to Exercises Chapter 4: Recurrence relations and generating functions Solutios to Exercises Chapter 4: Recurrece relatios ad geeratig fuctios 1 (a) There are seatig positios arraged i a lie. Prove that the umber of ways of choosig a subset of these positios, with o two chose

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 2006 Scoring Guidelines Form B AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

More information

5.3. Generalized Permutations and Combinations

5.3. Generalized Permutations and Combinations 53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

More information

The Field Q of Rational Numbers

The Field Q of Rational Numbers Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

More information

Multiplexers and Demultiplexers

Multiplexers and Demultiplexers I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

Simple Annuities Present Value.

Simple Annuities Present Value. Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

More information

Lecture 4: Cheeger s Inequality

Lecture 4: Cheeger s Inequality Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Section 8.3 : De Moivre s Theorem and Applications

Section 8.3 : De Moivre s Theorem and Applications The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

SEQUENCES AND SERIES CHAPTER

SEQUENCES AND SERIES CHAPTER CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

More information

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 0 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages, diagram sheet ad iformatio sheet. Please tur over Mathematics/P DBE/November 0

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

CHAPTER 11 Financial mathematics

CHAPTER 11 Financial mathematics CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Approximating the Sum of a Convergent Series

Approximating the Sum of a Convergent Series Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

More information

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau

More information

Figure 40.1. Figure 40.2

Figure 40.1. Figure 40.2 40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called

More information

Permutations, the Parity Theorem, and Determinants

Permutations, the Parity Theorem, and Determinants 1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

Chapter 6: CPU Scheduling. Previous Lectures. Basic Concepts. Histogram of CPU-burst Times. CPU Scheduler. Alternating Sequence of CPU And I/O Bursts

Chapter 6: CPU Scheduling. Previous Lectures. Basic Concepts. Histogram of CPU-burst Times. CPU Scheduler. Alternating Sequence of CPU And I/O Bursts Multithreadig Memory Layout Kerel vs User threads Represetatio i OS Previous Lectures Differece betwee thread ad process Thread schedulig Mappig betwee user ad kerel threads Multithreadig i Java Thread

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

Time Value of Money, NPV and IRR equation solving with the TI-86

Time Value of Money, NPV and IRR equation solving with the TI-86 Time Value of Moey NPV ad IRR Equatio Solvig with the TI-86 (may work with TI-85) (similar process works with TI-83, TI-83 Plus ad may work with TI-82) Time Value of Moey, NPV ad IRR equatio solvig with

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is

More information