. 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 Vocabulary sequece terms of a sequece series summatio otatio sigma otatio KEY CONCEPT Sequeces For Your Notebook A sequece is a fuctio whose domai is a set of cosecutive itegers. If a domai is ot specified, it is uderstood that the domai starts with. The values i the rage are called the terms of the sequece. Domai:... The relative positio of each term Rage: a a a a... a Terms of the sequece A fiite sequece has a limited umber of terms. A ifiite sequece cotiues without stoppig. Fiite sequece:,,, 8 Ifiite sequece:,,, 8,... A sequece ca be specified by a equatio, or rule. For example, both sequeces above ca be described by the rule a or f(). E XAMPLE Write terms of sequeces Write the first six terms of (a) a ad (b) f() (). a. a () 7 st term b. f() () st term a () 9 d term f() () d term a () rd term f() () 9 rd term a () th term f() () 7 th term a () th term f() () 8 th term a () 7 th term f() () th term GUIDED PRACTICE for Example Write the first six terms of the sequece.. a. f() (). a } 79 Chapter Sequeces ad Series
WRITING RULES If the terms of a sequece have a recogizable patter, the you may be able to write a rule for the th term of the sequece. E XAMPLE Write rules for sequeces WRITE RULES If you are give oly the first several terms of a sequece, there is o sigle rule for the th term. For istace, the sequece,, 8,... ca be give by a or a. Describe the patter, write the ext term, ad write a rule for the th term of the sequece (a), 8, 7,,... ad (b) 0,,,,.... a. You ca write the terms as (), (), (), (),.... The ext term is a (). A rule for the th term is a (). b. You ca write the terms as 0(), (), (), (),.... The ext term is f() () 0. A rule for the th term is f() ( ). GRAPHING SEQUENCES To graph a sequece, let the horizotal axis represet the positio umbers (the domai) ad the vertical axis represet the terms (the rage). E XAMPLE TAKS REASONING: Multi-Step Problem RETAIL DISPLAYS You work i a grocery store ad are stackig apples i the shape of a square pyramid with 7 layers. Write a rule for the umber of apples i each layer. The graph the sequece. STEP Make a table showig the umber of fruit i the first three layers. Let a represet the umber of apples i layer. Layer, Number of apples, a 9 AVOID ERRORS Although the plotted poits i Example follow a curve, do ot draw the curve because the sequece is defied oly for iteger values of. STEP Write a rule for the umber of apples i each layer. From the table, you ca see that a. STEP Plot the poits (, ), (, ), (, 9),..., (7, 9). The graph is show at the right. Number of apples a 8 0 0 Layer GUIDED PRACTICE for Examples ad. For the sequece, 8,,,..., describe the patter, write the ext term, graph the first five terms, ad write a rule for the th term.. WHAT IF? I Example, suppose there are 9 layers of apples. How may apples are i the 9th layer?. Defie ad Use Sequeces ad Series 79
KEY CONCEPT For Your Notebook Series ad Summatio Notatio Whe the terms of a sequece are added together, the resultig expressio is a series. A series ca be fiite or ifiite. Fiite series: 8 Ifiite series: 8... You ca use summatio otatio to write a series. For example, the two series above ca be writte i summatio otatio as follows: READING Whe writte i summatio otatio, this series is read as the sum of i for values of i from to. 8 i 8... ` For both series, the idex of summatio is i ad the lower limit of summatio is. The upper limit of summatio is for the fiite series ad ` (ifiity) for the ifiite series. Summatio otatio is also called sigma otatio because it uses the uppercase Greek letter sigma, writte S. i E XAMPLE Write series usig summatio otatio Write the series usig summatio otatio. a. 0 7... 0 b. } } } }... a. Notice that the first term is (), the secod is (), the third is (), ad the last is (0). So, the terms of the series ca be writte as: i where,,,..., 0 The lower limit of summatio is ad the upper limit of summatio is 0. c The summatio otatio for the series is 0 i. b. Notice that for each term the deomiator of the fractio is more tha the umerator. So, the terms of the series ca be writte as: i } i where,,,,... The lower limit of summatio is ad the upper limit of summatio is ifiity. i c The summatio otatio for the series is }. i ` GUIDED PRACTICE for Example Write the series usig summatio otatio.. 0... 00 7. } } 9 } 0 } 7... 8. 9... 9. 7... 79 Chapter Sequeces ad Series
INDEX OF SUMMATION The idex of summatio for a series does ot have to be i ay letter ca be used. Also, the idex does ot have to begi at. For istace, the idex begis at i the ext example. AVOID ERRORS Be sure to use the correct lower ad upper limits of summatio whe fidig the sum of a series. E XAMPLE Fid the sum of the series. Fid the sum of a series 8 ( k ) ( ) ( ) ( ) ( 7 ) ( 8 ) k 9 8 9 7 0 SPECIAL FORMULAS For series with may terms, fidig the sum by addig the terms ca be tedious. Below are formulas you ca use to fid the sums of three special types of series. KEY CONCEPT For Your Notebook Formulas for Special Series Sum of terms of Sum of first positive itegers Sum of squares of first positive itegers ( ) i } i ( )( ) } E XAMPLE Use a formula for a sum RETAIL DISPLAYS How may apples are i the stack i Example o page 79? From Example you kow that the ith term of the series is give by i where,,,..., 7. Usig summatio otatio ad the third formula listed above, you ca fid the total umber of apples as follows:... 7 7 i 7(7 )( p 7 ) } } 7(8)() 0 c There are 0 apples i the stack. Check this by actually addig the umber of apples i each of the seve layers. GUIDED PRACTICE for Examples ad Fid the sum of the series. 0. 8i. 7 k (k )... WHAT IF? Suppose there are 9 layers i the apple stack i Example. How may apples are i the stack?. Defie ad Use Sequeces ad Series 797
. EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS o p. WS for Exs. 9, 7, ad TAKS PRACTICE AND REASONING Exs. 7, 8,, 7, 9, ad 70. VOCABULARY Copy ad complete: Aother ame for summatio otatio is?.. WRITING Explai the differece betwee a sequece ad a series. EXAMPLE o p. 79 for Exs. WRITING TERMS Write the first six terms of the sequece.. a. a. a. f() 7. a 8. a 9. f() 0. a ( ). f() }. a }. a }. f() } EXAMPLE o p. 79 for Exs. 7 WRITING RULES For the sequece, describe the patter, write the ext term, ad write a rule for the th term..,,,,....,,, 8,... 7., 8,,,... 8., 9, 8,,... 9.... 0. 9 8........ 7.....,.8,.,.,... 0 0 0 0..,.,, 0.,.,.....,., 9.,.,.... 9,.8,.,.,... 7. TAKS REASONING Which rule gives the total umber of squares i the th figure of the patter show? A a B a C a D a ( ) } EXAMPLE o p. 79 for Exs. 8 GRAPHING SEQUENCES Graph the sequece. 8.,, 8,, 9.,, 8,,, 0.,, 9,,..., 9.,,, 8,...,. 0,, 8,,,., 0,, 8, 7., 9,, 9,...., }...., } 9 9 8 7 EXAMPLE o p. 79 for Exs. 7 WRITING SUMMATION NOTATION Write the series usig summatio otatio. 7. 7 0 9 8. 7 9 9. 7... 0. 8.... 0 7..... } } 9 } 7 } 8 } } } } 7 } 8 } 9 7 } 0. 7... 798 Chapter Sequeces ad Series
EXAMPLES ad o p. 797 for Exs. 8 USING SUMMATION NOTATION Fid the sum of the series.. 9. k i. (k ) 0. 7i 7. ( ). 0 8 8. }. i k k k k } k... i. 8 7. ERROR ANALYSIS Describe ad correct the error i fidig the sum of the series. i 0 (i ) 7 9 8. 0 TAKS REASONN I G What is the sum of the series i? A 0 B 0 C 0 D 870 REVIEW LOGIC For help with couterexamples see p. 00. CHALLENGE Tell whether the statemet about summatio otatio is true or false. If the statemet is true, prove it. If the statemet is false, give a couterexample. 9.. k k b i 0. b i. ( b i ) ( ) k k b i PROBLEM SOLVING EXAMPLES ad o pp. 79 797 for Exs.. GEOMETRY For a regular -sided polygo ( ), the measure a of a iterior agle is give by this formula: 80( ) a } Write the first five terms of the sequece. Write a rule for the sequece givig the total measure T of the iterior agles i each regular -sided polygo. Use the rule to fid the total measure of the agles i the Guggeheim Museum skylight, which is a regular dodecago. Guggeheim Museum Skylight. TAKS REASONING You wat to save $00 for a school trip. You begi by savig a pey o the first day. You pla to save a additioal pey each day after that. For example, you will save peies o the secod day, peies o the third day, ad so o. How much moey will you have saved after 00 days? How may days must you save to have saved $00? Explai how you used a series to fid your aswer.. Defie ad Use Sequeces ad Series 799
. TOWER OF HANOI I the puzzle called the Tower of Haoi, the object is to use a series of moves to take the rigs from oe peg ad stack them i order o aother peg. A move cosists of movig exactly oe rig, ad o rig may be placed o top of a smaller rig. The miimum umber a of moves required to move rigs is for rig, for rigs, 7 for rigs, for rigs, ad for rigs. Fid a formula for the sequece. What is the miimum umber of moves required to move rigs? 7 rigs? 8 rigs? Start Step Step Step Ed. MULTI-STEP PROBLEM The mea distace d (i astroomical uits) of each plaet (except Neptue) from the su is approximated by the Titius-Bode rule, d 0.() 0., where is a positive iteger represetig the positio of the plaet from the su. a. Evaluate The value of is for Mars. Use the Titius-Bode rule to approximate the distace of Mars from the su. b. Covert Oe astroomical uit is equal to about 9,00,000 kilometers. How far is Mars from the su i kilometers? c. Graph Graph the sequece give by the Titius-Bode rule. 7. TAKS REASONING For a display at a sports store, you are stackig soccer balls i a pyramid whose base is a equilateral triagle. The ( ) umber a of balls per layer is give by a } where represets the top layer. a. How may balls are i the fifth layer? b. How may balls are i a stack with five layers? c. Compare the umber of balls i a layer of a triagular pyramid with the umber of balls i the same layer of a square pyramid. 8. CHALLENGE Usig the true statemets from Exercises 9 o page 799 ad the special formulas o page 797, fid a formula for the umber of balls i the top layers of the pyramid from Exercise 7. MIXED REVIEW FOR TAKS TAKS PRACTICE at classzoe.com REVIEW Lesso.; TAKS Workbook 9. TAKS PRACTICE The sale price, y, for a pair of teis shoes is } of the origial price, x. Which equatio represets this relatioship? TAKS Obj. A y } x B y } x C y x } D y x } REVIEW Lesso.; TAKS Workbook 70. TAKS PRACTICE Which equatio represets a lie with a slope of ad a y-itercept of? TAKS Obj. F y x G y x 0 H y x J y x 0 800 EXTRA PRACTICE for Lesso., p. 0 ONLINE QUIZ at classzoe.com
Graphig Calculator ACTIVITY Use after Lesso.. Work with Sequeces TEKS a., a., a.; P..A TEXAS classzoe.com Keystrokes QUESTION How ca you use a graphig calculator to perform operatios with sequeces? EXAMPLE Fid, graph, ad sum terms of a sequece Use a graphig calculator to fid the first eight terms of a. Graph the sequece. The fid the sum of the first eight terms of the sequece. STEP Eter sequece Put the graphig calculator i sequece mode ad dot mode. Eter the sequece. Note that the calculator uses u() rather tha a. STEP Calculate terms Use the table feature to view the terms of the sequece. The first eight terms are, 7,, 7,, 7,, ad 7. Mi= u()=- u(mi)= v()= v(mi)= w()= w(mi)= = u() 7 7 STEP Graph sequece Set the viewig widow so that 8, 0 x 9, ad 0 y 0. Graph the sequece. Use the trace feature to view the terms of the sequece. STEP Fid sum of terms Use the summatio feature to fid the sum of the first eight terms of the sequece. The scree shows that the sum is. sum(seq(-,,, 8)) = X= Y= P RACTICE Use a graphig calculator to (a) fid the first te terms of the sequece, (b) graph the sequece, ad (c) fid the sum of the first te terms of the sequece.. a. a ( ). a. a. a. a. Defie ad Use Sequeces ad Series 80