T1.1 Lesson 3 - Arithmetic & Geometric Series & Summation Notation

Size: px
Start display at page:

Download "T1.1 Lesson 3 - Arithmetic & Geometric Series & Summation Notation"

Transcription

1 Fast Five T. Lesso 3 - Arithmetic & Geometric eries & ummatio Notatio Math L - atowski Fid the sum of the first 00 umbers Outlie a way to solve this problem ad the carry out your pla Fid the sum of the first 25 perfect squares Outlie a way to solve this problem ad the carry out your pla 8//2009 Math L - atowski 8//2009 T. - equeces & eries - Lesso 2 (A) Review (B) Arithmetic eries A sequece is a set of ordered terms, possibly related by some patter Oe such patter is called arithmetic because each pair of cosecutive terms has a commo differece The geeral term of a arithmetic sequece is defied by the formula u u + ( - )d A geometric sequece is oe i which the cosecutive terms differ by a commo ratio The geeral term of a geometric sequece is defied by the formula u u x r ( - ) A series is defied as the sum of the terms of a sequece. As a example, start by fidig the sum of the first 00 umbers ad showig a easy way to set it up: o the the sum is (0)(00) 2 8//2009 Math L - atowski 3 8//2009 Math L - atowski.0. - Arithmetic equeces

2 (B) Arithmetic eries (C) Examples For a arithmetic sequece the the formula for the sum of its terms is: ( a + t) [2 a + ( ) d] 2 2 Ex. Fid the sum of the series Ex 2. For the series , fid u 20 ad 20 Ex 3. The fifth term of a arithmetic series is 9 ad the sum of the first is 80. Fid the first three terms of the series. Ex. I a arithmetic series of 50 terms, the 7th term is 53 ad the 28th term is 8. Fid the sum of the series. 8//2009 Math L - atowski 5 8//2009 Math L - atowski (C) Examples (D) ummatio Notatio ex 5. hayla deposits $28 ito her accout. Each week she deposits $7 less tha the previous week util she deposits her last deposit of $2. What total amout did she deposit? ex. Jaye buys 0 widgets o the Ja st, 5 o the st of Feb, 20 o the st of March, etc... How may widgets has she acquired i 2 years? How log does it take her to acquire 5,000 widgets? ummatio otatio is a shorthad way of sayig take the sum of certai terms of a sequece the Greek letter sigma, Σ is used to idicate a summatio I the expressio i represets the term umber (or idex of summatio), ad a i represets the geeral term of the sequece beig summed o therefore, a i a a + a + a + a a i 2 3 8//2009 Math L - atowski 7 8//2009 Math L - atowski Arithmetic equeces

3 (E) ummatio Notatio Lesso 2 Fast Five Ex Arithmetic eries ( + ) ( + ) + (2 + ) + (3 + ) + ( + ) + (5 + ) + ( + ) ( + ) (2) + (3) + () + (5) + () + (7) ( + ) 27 Determie NO CALCULATOR 8//2009 Math L - atowski 9 (E) Examples of ummatio Notatio (E) Examples of ummatio Notatio Write out the series expasio for the followig ad the use the TI-8 to evaluate the sums: i 2 i i e l ( i) Evaluate the series that are defied i the followig summatio otatios: ( 3i 5) + 20 ( 5) 0 25 i 2 ( i 5) ( i) ( i) is 8//2009 Math L - atowski Arithmetic equeces

4 (F) Geometric eries (F) Geometric eries Fid the sum of the first 7 terms of the series a + ar + ar 2 + ar 3 + ar + ar 5 + ar ( ) - ( ) o i geeral, the formula for the sum of a geometric series is: ( u+ u) u ( r ) r r 7 ½(3 7 - ) 8//2009 Math L - atowski 3 8//2009 Math L - atowski (G) Examples (G) Examples ex. Fid 8 give ad rewrite i summatio otatio: (a) (b) ex 2. Fid the total amout you make if you were paid a rupee a day, but the amout was doubled every day for a moth ex 3. Fid the sum / + / Ex. The fifth term of a geometric series is 05 ad the sixth term is 25. Fid the sum of the first ie terms. ex 5. A ball drops from a height of m ad its height o the bouce is 5/8th of the previous maximum height. Determie the total height bouced by the ball after it touches the groud for the 7 th bouce. 8//2009 Math L - atowski 5 8//2009 Math L - atowski.0. - Arithmetic equeces

5 (H) Examples of ummatio Notatio (H) Examples of ummatio Notatio Ex 2 Geometric eries Give the series ( ) ()( 0.5) + ()( 0.5) + ()( 0.5) + ()( 0.5) ( ) ()() + ()( 0.5) + ()(0.25) + ()( 0.25) 2 ()( ) + ( 2) + + ( 0.5) (i) Write a summatio expressio for the series (ii) Determie (iii) Determie 5 (iv) Determie 2 (v) Predict,000,000 8//2009 Math L - atowski 7 8//2009 Math L - atowski 8 (H) Examples of ummatio Notatio (H) Examples of ummatio Notatio The series ½ + ¼ + /8 + / +... is a example of a ifiite geometric series. (a) Determie the sum of this series. (b) Is it possible to fid the sum of ay ifiite geometric sequece? Explai. (c) Uder what coditios is it possible to fid the sum of a ifiite geometric sequece how that the sum of terms of the series ½ + ¼ +... u is always less tha, where is ay atural umber. Explai what the followig otatios mea: lim 2 i ( 2 ) 2 i ( 2 ) 2 i ( 2 ) 8//2009 Math L - atowski 9 8//2009 Math L - atowski Arithmetic equeces

6 (I) Iteret Liks (J) Homework Geometric equeces & eries From West Texas A&M Arithmetic equeces & eries From West Texas A&M U HW: Ex 2E. #ae; Ex 2E.2 #c, 2a, 3,5,, ; Ex 2E.3 #bc, 2cd,,, 7 HW Ex 2F #c, 3c, c, 5ab ad IB packet # - 8 8//2009 Math L - atowski 2 8//2009 Math L - atowski Arithmetic equeces

Unit 6: Sequences and Series

Unit 6: Sequences and Series AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo

More information

Section 6.4: Series. Section 6.4 Series 413

Section 6.4: Series. Section 6.4 Series 413 ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they

More information

is also known as the general term of the sequence

is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

More information

CHAPTER 2 SEQUENCES AND SERIES OUTLINE Geometric Series Infinite Geometric Series and Sigma Notation 4 Review

CHAPTER 2 SEQUENCES AND SERIES OUTLINE Geometric Series Infinite Geometric Series and Sigma Notation 4 Review CHAPTER SEQUENCES AND SERIES OUTLINE Day Sectio Topic.7 Geometric Sequeces.8 Geometric Series 3.9 Ifiite Geometric Series ad Sigma Notatio 4 Review 5 Review 6 Chapter Test Terry Fox Math 007 .7 GEOMETRIC

More information

Section 7 Fundamentals of Sequences and Series

Section 7 Fundamentals of Sequences and Series ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which

More information

11.1 Arithmetic Sequences and Series

11.1 Arithmetic Sequences and Series 11.1 Arithmetic Sequeces ad Series A itroductio 1, 4, 7, 10, 13 9, 1, 7, 15 6., 6.6, 7, 7.4 ππ+, 3, π+ 6 Arithmetic Sequeces ADD To get ext term 35 1 7. 3π + 9, 4, 8, 16, 3 9, 3, 1, 1/ 3 1,1/ 4,1/16,1/

More information

MIXED REVIEW of Problem Solving

MIXED REVIEW of Problem Solving MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTI-STEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous

More information

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio

More information

Define and Use Sequences and Series

Define and Use Sequences and Series . a., A..A; P..A, P..B TEKS Defie ad Use Sequeces ad Series Before You idetified ad wrote fuctios. Now You will recogize ad write rules for umber patters. Why? So you ca fid agle measures, as i Ex.. Key

More information

Sequences, Sums, and Products

Sequences, Sums, and Products CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet

More information

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE

More information

If we want to add up the area of four rectangles, we could find the area of each rectangle and then write this sum symbolically as:

If we want to add up the area of four rectangles, we could find the area of each rectangle and then write this sum symbolically as: Sigma Notatio: If we wat to add up the area of four rectagles, we could fid the area of each rectagle ad the write this sum symbolically as: Sum A A A A Liewise, the sum of the areas of te triagles could

More information

2.4 - Sequences and Series

2.4 - Sequences and Series 2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,

More information

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad

More information

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018 CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical

More information

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) VERY SHORT ANSWER TYPE QUESTIONS ( MARK). If th term of a A.P. is 6 7 the write its 50 th term.. If S = +, the write a. Which term of the sequece,, 0, 7,... is 6? 4. If i a A.P. 7 th term is 9 ad 9 th

More information

Lecture 2.5: Sequences

Lecture 2.5: Sequences Lecture.5: Sequeces CS 50, Discrete Structures, Fall 015 Nitesh Saxea Adopted from previous lectures by Zeph Gruschlag Course Admi HW posted Covers Chapter Due Oct 0 (Tue) Mid Term 1: Oct 15 (Thursday)

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Binomial distribution questions: formal word problems

Binomial distribution questions: formal word problems Biomial distributio questios: formal word problems For the followig questios, write the iformatio give i a formal way before solvig the problem, somethig like: X = umber of... out of 2, so X B(2, 0.2).

More information

BC: Q401.CH9A Convergent and Divergent Series (LESSON 1)

BC: Q401.CH9A Convergent and Divergent Series (LESSON 1) BC: Q40.CH9A Coverget ad Diverget Series (LESSON ) INTRODUCTION Sequece Notatio: a, a 3, a,, a, Defiitio: A sequece is a fuctio f whose domai is the set of positive itegers. Defiitio: A ifiite series (or

More information

AP CALCULUS - AB LECTURE NOTES MS. RUSSELL

AP CALCULUS - AB LECTURE NOTES MS. RUSSELL AP CALCULUS - AB LECTURE NOTES MS. RUSSELL Sectio Number: 4. Topics: Area -Sigma Notatio Part: of Sigma Notatio Upper boud Recall ai = a+ a + a3 + L + a idex i= Lower boud Example : Evaluate each summatio.

More information

Chapter 7: Numerical Series

Chapter 7: Numerical Series Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

Sigma notation. 2.1 Introduction

Sigma notation. 2.1 Introduction Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate

More information

ARITHMETIC PROGRESSIONS

ARITHMETIC PROGRESSIONS CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first

More information

Infinite Series. Definition. An infinite series is an expression of the form. Where the numbers u k are called the terms of the series.

Infinite Series. Definition. An infinite series is an expression of the form. Where the numbers u k are called the terms of the series. Ifiite Series Defiitio. A ifiite series is a expressio of the form uk = u + u + u + + u + () 2 3 k Where the umbers u k are called the terms of the series. Such a expressio is meat to be the result of

More information

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics: Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets

More information

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play. Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers

More information

XT - MATHS Grade 12. Date: 2010/06/29. Subject: Series and Sequences 1: Arithmetic Total Marks: 84 = 2 = 2 1. FALSE 10.

XT - MATHS Grade 12. Date: 2010/06/29. Subject: Series and Sequences 1: Arithmetic Total Marks: 84 = 2 = 2 1. FALSE 10. ubject: eries ad equeces 1: Arithmetic otal Mars: 8 X - MAH Grade 1 Date: 010/0/ 1. FALE 10 Explaatio: his series is arithmetic as d 1 ad d 15 1 he sum of a arithmetic series is give by [ a ( ] a represets

More information

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6 Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is

More information

a n b. n a n Test Review for Lessons 9.1 and 9.3 Show all work on a separate sheet of paper for full credit.

a n b. n a n Test Review for Lessons 9.1 and 9.3 Show all work on a separate sheet of paper for full credit. Precalculus Test Review for Lessos. ad. Name: Date: Hour: Show all work o a separate sheet of paper for full credit.. Write the first four terms of the sequeces give by a. a b. a 7. Write the first five

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

Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1

Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1 . Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely

More information

= n which will be written with general term a n Represent this sequence by listing its function values in order:

= n which will be written with general term a n Represent this sequence by listing its function values in order: Sectio 9.: Sequeces a Series (Sums) Remier:, 2, 3,, k 2, k, k, k +, k + 2, The ellipsis ots iicate the sequece is oeig, has ifiitely may terms. I. SEQUENCES Defiitio (page 706): A sequece is a fuctio whose

More information

CHAPTER - 9 SEQUENCES AND SERIES KEY POINTS A sequece is a fuctio whose domai is the set N of atural umbers. A sequece whose rage is a subset of R is called a real sequece. Geeral A.P. is, a, a + d, a

More information

Lesson 1.1 Recursively Defined Sequences

Lesson 1.1 Recursively Defined Sequences Lesso 1.1 Recursively Defied Sequeces 1. Fid the commo differece, d, for each arithmetic sequece ad the commo ratio, r, for each geometric sequece. a. 1., 1.,.,,.,... b..62,.12,.2,... c. 1,.2,.4,.8,...

More information

ARITHMETIC PROGRESSION

ARITHMETIC PROGRESSION CHAPTER 5 ARITHMETIC PROGRESSION Poits to Remember :. A sequece is a arragemet of umbers or objects i a defiite order.. A sequece a, a, a 3,..., a,... is called a Arithmetic Progressio (A.P) if there exists

More information

Roberto s Notes on Infinite Series Chapter 1: Sequences and series Section 3. Geometric series

Roberto s Notes on Infinite Series Chapter 1: Sequences and series Section 3. Geometric series Roberto s Notes o Ifiite Series Chapter 1: Sequeces ad series Sectio Geometric series What you eed to kow already: What a ifiite series is. The divergece test. What you ca le here: Everythig there is to

More information

Starting with a time of 1 second, the partial sums of the time series form the sequence 1, 3 2, 7 4, 1 5

Starting with a time of 1 second, the partial sums of the time series form the sequence 1, 3 2, 7 4, 1 5 - OBJECTIVE Determie whether a series is coverget or diverget. Coverget ad Diverget Series HISTORY The Greek philosopher Zeo of Elea (c. 90 30 B.C.) proposed several perplexig riddles, or paradoxes. Oe

More information

SEQUENCE AND SERIES NCERT

SEQUENCE AND SERIES NCERT 9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of

More information

Progressions. ILLUSTRATION 1 11, 7, 3, -1, i s an A.P. whose first term is 11 and the common difference 7-11=-4.

Progressions. ILLUSTRATION 1 11, 7, 3, -1, i s an A.P. whose first term is 11 and the common difference 7-11=-4. Progressios SEQUENCE A sequece is a fuctio whose domai is the set N of atural umbers. REAL SEQUENCE A Sequece whose rage is a subset of R is called a real sequece. I other words, a real sequece is a fuctio

More information

Objective Mathematics

Objective Mathematics . If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic

More information

Further Exploration of Patterns

Further Exploration of Patterns Further Exploratio of Patters Abstract Quadratic Patters a+b+c 4a+b+c 9a+3b+c 16a+4b+c 5a+5b+c 1 st chage 3a+b 5a+b 7a+b 9a+b d chage a a a 1 Coefficiet of is ( a) a. Costat secod chage, therefore it is

More information

Created by T. Madas SERIES. Created by T. Madas

Created by T. Madas SERIES. Created by T. Madas SERIES SUMMATIONS BY STANDARD RESULTS Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, 66638 Questio (**+) Fid, i fully simplified factorized form, a expressio

More information

Chapter 6. Progressions

Chapter 6. Progressions Chapter 6 Progressios Evidece is foud that Babyloias some 400 years ago, kew of arithmetic ad geometric progressios. Amog the Idia mathematicias, Aryabhata (470 AD) was the first to give formula for the

More information

Section 5.5. Infinite Series: The Ratio Test

Section 5.5. Infinite Series: The Ratio Test Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

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

Essential Question How can you recognize an arithmetic sequence from its graph?

Essential Question How can you recognize an arithmetic sequence from its graph? . Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the

More information

Basic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.

Basic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S. Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

Math 2412 Review 3(answers) kt

Math 2412 Review 3(answers) kt Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled

More information

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3. Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

10.2 Infinite Series Contemporary Calculus 1

10.2 Infinite Series Contemporary Calculus 1 10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,

More information

= = =

= = = Sec 5.8 Sec 6. Mathematical Modelig (Arithmetic & Geometric Series) Name: Carl Friedrich Gauss is probably oe of the most oted complete mathematicias i history. As the story goes, he was potetially recogiized

More information

GRADE 12 JUNE 2016 MATHEMATICS P1

GRADE 12 JUNE 2016 MATHEMATICS P1 NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 016 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 14 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/JUNE 016) INSTRUCTIONS

More information

GRADE 12 SEPTEMBER 2015 MATHEMATICS P1

GRADE 12 SEPTEMBER 2015 MATHEMATICS P1 NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 015 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 10 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/SEPTEMBER 015) INSTRUCTIONS

More information

Solutions to Math 347 Practice Problems for the final

Solutions to Math 347 Practice Problems for the final Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is

More information

Lesson 1.1 Recursively Defined Sequences

Lesson 1.1 Recursively Defined Sequences Lesso 1.1 Recursively Defied Sequeces 1. Tell whether each sequece is arithmetic, geometric, or either. a. 1,, 9, 13,... b. 2, 6, 18, 4,... c. 1, 1, 2, 3,, 8,... d. 16, 4, 1,.2,... e. 1, 1, 1, 1,... f..6,

More information

Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions

Precalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group

More information

Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of

Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of Brai Teasures Progressio ad Series By Abhijit kumar Jha EXERCISE I Q If the 0th term of a HP is & st term of the same HP is 0, the fid the 0 th term Q ( ) Show that l (4 36 08 up to terms) = l + l 3 Q3

More information

Areas and Distances. We can easily find areas of certain geometric figures using well-known formulas:

Areas and Distances. We can easily find areas of certain geometric figures using well-known formulas: Areas ad Distaces We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate the area of the regio

More information

n=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n

n=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9. Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say that

More information

Name Date PRECALCULUS SUMMER PACKET

Name Date PRECALCULUS SUMMER PACKET Name Date PRECALCULUS SUMMER PACKET This packet covers some of the cocepts that you eed to e familiar with i order to e successful i Precalculus. This summer packet is due o the first day of school! Make

More information

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n.

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n. COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where

More information

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

Quiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit

Quiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE

More information

Roberto s Notes on Series Chapter 2: Convergence tests Section 7. Alternating series

Roberto s Notes on Series Chapter 2: Convergence tests Section 7. Alternating series Roberto s Notes o Series Chapter 2: Covergece tests Sectio 7 Alteratig series What you eed to kow already: All basic covergece tests for evetually positive series. What you ca lear here: A test for series

More information

4.1 SIGMA NOTATION AND RIEMANN SUMS

4.1 SIGMA NOTATION AND RIEMANN SUMS .1 Sigma Notatio ad Riema Sums Cotemporary Calculus 1.1 SIGMA NOTATION AND RIEMANN SUMS Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each

More information

HUMBEHV 3HB3 Measures of Central Tendency & Variability Week 2

HUMBEHV 3HB3 Measures of Central Tendency & Variability Week 2 Describig Data Distributios HUMBEHV 3HB3 Measures of Cetral Tedecy & Variability Week 2 Prof. Patrick Beett Ofte we wish to summarize distributios of data, rather tha showig histograms Two basic descriptios

More information

Exponential and Trigonometric Functions Lesson #1

Exponential and Trigonometric Functions Lesson #1 Epoetial ad Trigoometric Fuctios Lesso # Itroductio To Epoetial Fuctios Cosider a populatio of 00 mice which is growig uder plague coditios. If the mouse populatio doubles each week we ca costruct a table

More information

Sequences. Notation. Convergence of a Sequence

Sequences. Notation. Convergence of a Sequence Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it

More information

multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2.

multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2. Lesso 3- Lesso 3- Scale Chages of Data Vocabulary scale chage of a data set scale factor scale image BIG IDEA Multiplyig every umber i a data set by k multiplies all measures of ceter ad the stadard deviatio

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Essential Question How can you use properties of exponents to simplify products and quotients of radicals?

Essential Question How can you use properties of exponents to simplify products and quotients of radicals? . TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.7.G Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties

More information

ENGI Series Page 6-01

ENGI Series Page 6-01 ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,

More information

MATH 2300 review problems for Exam 2

MATH 2300 review problems for Exam 2 MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. (a) Fid the mass of the plate. Iclude uits. Mass =

More information

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1 Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits

More information

Chapter 6: Numerical Series

Chapter 6: Numerical Series Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

Math 116 Second Midterm November 13, 2017

Math 116 Second Midterm November 13, 2017 Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.

More information

Mathematics: Paper 1

Mathematics: Paper 1 GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists

More information

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function. MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n

07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n 07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( ) If = a, y = b, z = c, where a, b, c are i A.P. ad = 0 = 0 = 0 l a l

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

Infinite Series and Improper Integrals

Infinite Series and Improper Integrals 8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to

More information

} is said to be a Cauchy sequence provided the following condition is true.

} is said to be a Cauchy sequence provided the following condition is true. Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r

More information

Taylor Series (BC Only)

Taylor Series (BC Only) Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

Solving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots)

Solving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots) Evet A: Fuctios ad Algebraic Maipulatio Factorig Square of a sum: ( a + b) = a + ab + b Square of a differece: ( a b) = a ab + b Differece of squares: a b = ( a b )(a + b ) Differece of cubes: a 3 b 3

More information

CS161 Handout 05 Summer 2013 July 10, 2013 Mathematical Terms and Identities

CS161 Handout 05 Summer 2013 July 10, 2013 Mathematical Terms and Identities CS161 Hadout 05 Summer 2013 July 10, 2013 Mathematical Terms ad Idetities Thaks to Ady Nguye ad Julie Tibshirai for their advice o this hadout. This hadout covers mathematical otatio ad idetities that

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

Sums of powers of the natural numbers

Sums of powers of the natural numbers Sums of powers of the atural umbers Dr Richard Kederdie Kederdie Maths Tutorig December 04 A commo questio asked of studets of mathematical iductio is to prove + +..... + = ( + ) ( + ). This ote derives

More information

10.6 ALTERNATING SERIES

10.6 ALTERNATING SERIES 0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose

More information

MATH 2300 review problems for Exam 2

MATH 2300 review problems for Exam 2 MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. Fid the mass of the plate. Iclude uits. Fid the ceter

More information