is also known as the general term of the sequence

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1 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 whether a sequece is arithmetic or geometric. I ca determie the geeral terms of a arithmetic ad geometric sequece. I ca determie the sum of a fiite arithmetic or geometric series. I ca determie the sum of certai ifiite geometric series. I ca use ad iterpret summatio otatio. Defiitios / Vocabulary / Graphical Iterpretatio: Sequece characteristics: List of umbers writte i a defiite order Ca be fiite or ifiite Does ot have to have a patter A fuctio whose domai is the set of positive itegers a is also kow as the geeral term of the sequece List some examples of sequeces: The Fiboacci Sequece: How to create the Fiboacci Sequece: {,,, 3, 5, 8, 3,.} by addig the previous two terms A recursively defied sequece is a sequece where the geeral term is related to the previous terms. Covergig Sequeces: To determie whether or ot a sequece coverges we ca look at what happes to the geeral term as gets ifiitely large. 63

2 Examples: a coverges to: 4 a coverges to: 3 7 a coverges to: a e coverges to: 45 a e coverges to: Bouded Sequeces: If all the terms of a sequece lie betwee umbers, say M ad N, such that the terms are ever greater tha M or less tha N, we say the sequece is bouded. A coverget sequece is bouded. However, a bouded sequece is ot ecessarily coverget. (A bouded sequece does ot ecessarily have a limit.) If a sequece is mootoe (always icreasig or always decreasig) ad bouded, it coverges. Arithmetic Sequeces: A sequece that has a costat amout of icrease or decrease betwee terms (the commo differece) is called a arithmetic sequece. The equatio for the geeral term of a arithmetic sequece is just a liear fuctio with slope d (represetig the commo differece). a a ( ) d Geometric Sequeces: A sequece that has a costat percet icrease or decrease i terms (commo ratio) is called a geometric sequece. The equatio for the geeral term of a geometric sequece is a expoetial fuctio with base r (represetig the commo ratio). a ar 64

3 Examples: ( ) 3 a 0 4 a) Write out the first 5 terms b) Is the sequece arithmetic, geometric, or either? a 0, a a for a) Write out the first 5 terms b) Is the sequece arithmetic, geometric, or either? c) Fid a explicit formula for a Series ad Summatio Notatio: Summatio otatio is a short-had way of deotig the sum of the terms of a sequece. As with sequeces, series ca be fiite or ifiite. a k Give a sequece { } from m to p of the sequece { a } is writte: ad umbers m ad p satisfyig k m p, the summatio p m where is called the idex of summatio m is the lower limit of summatio p is the upper limit of summatio a a a... a m m p 65

4 Properties of Summatio Notatio: Calculatig a Fiite Arithmetic Series Sum: S ( a a where is the umber of terms; a is the first term, ad a is the last term. Calculatig a Fiite Geometric Series Sum: All we eed to calculate a fiite geometric series is the first term a, the commo ratio r, ad the umber of terms. i a( r ) S a ar ar... ar ar, r r Calculatig a Ifiite Geometric Series Sum: Whe the commo ratio is betwee - ad, we ca fid the sum of a ifiite geometric series with the followig: i0 ) S Example: Fid the sum i a a ar ar ar ar r i i i. 49 k0 (3k 5) 4. i 0 3 i 66

5 Sequeces ad Series Activity Objectives: Use ad iterpret otatio for sequeces ad series Idetify arithmetic ad geometric sequeces Fid a formula for the th term of a sequece (whe possible) Use ad iterpret summatio otatio Fid the sum of fiite arithmetic ad geometric sequeces Fid the sum of ifiite geometric series (whe possible) 67

6 Notatio for Sequeces. Write out the first five terms of the followig sequeces: a. a b. a 4 c. a a a, 3 d. a a a a a,,. Idetify a4 i the followig sequeces: a., 7,, 7,, 7, 3,... b., 8, 4, 7, 6, 648,... c., b+, b+, 3b+, 4b+,... 68

7 Idetifyig Sequeces. Decide whether each sequece i the table is arithmetic, geometric, or either. Fid the geeral term, if possible. commi differece? costat ratio? type of sequece Geeral Term a., 7,, 7,, 7, 3,... b., 8, 4, 7, 6, 648,... c. 5,, 5,, 5,, 5,, 5,... d. /3, 7/3, 3/3, 9/3, 5/3,... e. 4, -,, -0.5, 0.5,... f., b+, b+, 3b+, 4b+,... g. 5, 0, 5, 0, 5, 0, -5, -0,... h., -,, -,, -,, -,,... i.,, 3, 4, 5,. Make up your ow example of a sequece that is: arithmetic: geometric: either: 69

8 Summatio Notatio. Expad ad calculate the sum. a. 4 i 3i b. 5 j0 3 j c. 7 ( ) k k k d. Make up your ow expressio usig sigma otatio ad the expad it.. Use sigma otatio to express each sum. a b c d. Make up your ow sum of terms ad the re-write it usig sigma otatio 70

9 Arithmetic Series The sum of terms of a arithmetic sequece: S ( a a ). Fid the sums of the followig fiite arithmetic series: a b. 400 i (3i ) c. ( p ) ( p ) ( 4 p )... ( 96 p ) 7 d. 4. A theater is costructed so that each row has 4 more seats tha the oe i frot of it. The first row has 0 seats ad there are 50 rows i the theater. a. How may seats i the 50 th row? b. How may total seats i the theater? 7

10 Fiite Geometric Series a r The sum of terms of a fiite geometric sequece: S ( r applies if r. Fid the sums of the followig fiite geometric series: a ). This formula oly b. 8 c. 0 i0 5(.) i d. 0 i0 5(0.) i 7

11 Ifiite Geometric Series The sum of a ifiite umber of terms of a geometric sequece: oly applies if - < r <.. Fid the sums of the followig ifiite geometric series: a S. This formula a r b. 5(0.) x0 x c. 5(0.) x x. A rubber ball is dropped from a height of 4 feet. Each time it bouces it goes half as high as before (i.e. after it hits it bouces up feet, tha foot ad so o). What is the total distace covered by the ball oce it is doe boucig? 73

12 Applicatio of a Geometric Series. A clothig outlet foud that whe they itroduced a ew shirt, it sold quickly at first, but the as time wet o they sold less ad less quatities of the desig each moth. For a particular shirt, they foud that they sold 300 uits i the first moth. They the foud their sales of the shirt dropped by roughly 5% each moth there after. a. How may shirts did they sell i the first 3 moths (total) after the shirt was itroduced? b. How may shirts did they sell i the first year after the shirt was itroduced? c. Based o their estimatio, how may shirts should the compay expect to sell total? Hit: Look at the summatio as time gets ifiitely large! 74

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