Section 3.2: Arithmetic Sequences and Series

Transcription

1 Sectio 3.: Arithmetic Sequeces Series Arithmetic Sequeces Let s strt out with efiitio: rithmetic sequece: sequece i which the ext term is fou by ig costt (the commo ifferece ) to the previous term Here re some exmples of rithmetic sequeces: ) 7,, 5, 9, b), 4, 3, 0, 59 c),.3,.6,.9, The first oe hs commo ifferece of 4, the seco 7, the thir 0.3. Note tht i ech of them, we c fi the commo ifferece by tkig y term subtrctig the previous term from it. For the followig sequeces, stte whether ech of them is rithmetic. ) 3, 0, 7, 4, b) 4, 5, 7, 0, c), 4, 8, 6, ),,,, Aswer ) Yes, becuse the commo ifferece is 7. b) No, becuse you re ot ig the sme umber ech time. c) No, becuse you re multiplyig by to get the ext term, ot ig. ) No, becuse the ifferece betwee ech pir of terms is ifferet. Agi, you c efie rithmetic sequece i oe of three wys: by listig the terms, by givig recursive efiitio, or by givig geerl efiitio.

2 Recursive Defiitios for Arithmetic Sequeces Let s look first t exmple. Give recursive efiitio for the sequece, 0, 8, 6, Aswer Recll tht recursive efiitio hs two prts: listig the first term givig the ptter. I this cse, the ptter is ig = 8 to the previous term to get the ext term. The recursive efiitio is therefore 8 More geerlly, the recursive formul for y rithmetic sequece is Geerl Formule for Arithmetic Sequeces isert vlue here> Let s exmie the previous exmple i more etil to see if we c recogize y ptters come up with geerl formul. Rewritig ech term, we get, 0, 8, 6,...,6,6,63,... So the 3 r term equls the first plus 6 times, the 4 th term equls the first plus 6 times 3, the th term will equl the first plus 6 times ( ). More geerlly, the th term will equl the first plus times ( ). I other wors, for y rithmetic sequece. Write geerl formul for the sequece, 0, 8, 6, Aswer This sequece is rithmetic with the first term commo ifferece 8.

3 The geerl formul is the tht = 8 6. Wht is the 50 th term i the sequece i the sequece, 0, 8, 6,? Aswer This is the sme sequece from the previous exmple. We my the use the formul we erive, = 8 6, with = The 50 th term is 394. Wht is the commo ifferece i the rithmetic sequece i which the first term is 8 the twelfth term is 59? Aswer The commo ifferece is 7. Which term hs vlue of 404 i the sequece 37, 8, 9,? Aswer

4 So is 37 is +9. The we wt to fi the vlue of for which equls The fiftieth term is 404. Fi the first four terms of the rithmetic sequece i which the thirteeth term is 97 the fiftieth term is 393. Aswer So 3 = = 393. The we fi tht However, this hs two ukows,. Let s look t 50 : We ow hve two ukows, but two equtios, givig us the system Solvig this system, we first multiply the top equtio by egtive :

5 A the the two equtios together, so tht the terms ccel out Now we substitute ito oe of the origil equtios: Sice = = 8, our sequece is the, 9, 7, 5, Arithmetic Series Recll tht S is the sum of the first terms of series. Let s look t couple of exmples of rithmetic series to see if we c ietify y ptters. Suppose we wish to tke some prtil sums of the series Let s first clculte S 6. We coul just fi the first six terms them up, but otice the followig: S 6 = The sum of the first lst umbers is 44. The sum of the seco seco-to-lst is lso 44. So is the sum of the thir thir-lst. So whe you tke the terms i pirs, ech pir hs the sme sum,, there re / pirs i totl. The S. Wht if, however, there re o umber of terms? Let s lso clculte S 7 : S 7 =

6 The sum of the first lst is 5, s is the sum of the ech ier pir. Notice tht the mile, upire vlue, is ½ of 5. So i sese, the mile term is ½ of pir, for totl of 3½ pirs. But tht s just 7/, which is our / i the origil formul! So we re still goo. The reltioship still works, for both o eve vlues of. S Fi the sum of the first forty terms of the series Aswer This is just the sme sequece s before, with = = 8. I orer to use our previous formul, however, we ee to clculte 40 before we c clculte S So, S 40 S The sum of the first forty terms is 630. (Much esier th writig out the first forty terms ig them up!) I the previous exmple, we use the formul for to clculte the lst term put its vlue ito the formul for S. We coul o tht i more geerl wy: S the lst expressio, which gives S s fuctio of the first term, the umber of terms, the commo ifferece, c lso be use to evlute series.

7 Fi the sum of the first oe hure terms of the sequece 5, 6, 7, 6,. Aswer This sum will just be , with = 5 =. S 00 S Clculte j3 Aswer The first term will be for j=3 will equl 5(3)+0=5. Next is j=4 will equl 5(4)+0=30, j=5 equlig 5(5)=35, so o. The lst term will be for j=8 will equl 5(8)+0=00. I other wors, our series is Is it rithmetic? Yes, with commo ifferece = 5. Wht else o we ee for our clcultio? The umber of terms equls (lst first +), so is (8-3+)=6. The S 6 S Pt the mth istructor sks her stuets to o five wor problems the first week, six the seco week, seve the thir week, so o, icresig the umber of wor problems ech week by oe. ) How my wor problems will iliget stuets be oig i the lst week of clsses (the th week)?

8 b) How my wor problems will iliget stuets hve complete urig the course of the term ( weeks)? Aswer ) The umber of wor problems is sequece: 5, 6, 7,. I fct, it s rithmetic sequece with = 5 =. I the eleveth week, the, 505 Diliget stuets will solve 5 wor problems i the lst week of clsses. b) The totl umber of wor problems solve is S S Diliget stuets will hve solve 0 wor problems i totl. Summry For rithmetic sequece, the th term is give by or For rithmetic series, the sum of the first terms (th prtil sum) is S S