Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic series d fid their sums. Vocbulry Sequece, term of sequece, explicit formul, recursive formul. Arithmetic sequece, commo differece. Arithmetic series,
Wht A sequece: is sequece? is ordered list of umbers. Ech umber i sequece is term of sequece.,,,...,, 3 st term d term - term th term + term A explicit formul describes the th term of sequece usig the umber. It is used to represet terms or fid term. For exmple:,4,6,8,0,.... The th term is twice the vlue of. A recursive formul: formul tht describes the th term of sequece by referrig to precedig terms. For exmple: 33,30,7,4,... 33 iitil coditio 3, for
Problem : Geertig Sequece usig o Explicit Formul A sequece hs 0 terms of this sequece? explicit formul 3. Wht re the first 3 3 3 4 Aswer:,4,7,0,3,6,9,,5,8. Your tur 3. Wht is the Aswer: 47 term i the sequece?
Problem : The umber of blocks i two-dimesiol pyrmid is sequece tht follows recursive formul. Wht is recursive defiitio for the sequece? Aswer Subtrct cosecutive terms to fid out wht hppes from oe term to the ext. 3 3 6 3 3 4 0 6 3 4 Cout the umber of blocks i ech pyrmid.,3,6,0, 5, Use to express the reltioship betwee successive terms. To write recursive defiitio, stte the iitil coditio d the recursive formul. d
Problem 3: Usig G.C. Usig Formuls to fid terms of sequece. Pierre beg the yer with upid of $300 o his credit crd. Becuse he hd ot red the credit crd greemet, he did ot relize tht the compy chrged.8% iterest ech moth o his upid blce, i dditio to $9 pelty i y moth he might fil to mke miimum pymet. Pierre igored his credit crd bill for 4 cosecutives moths before filly decidig to py off the blce. Wht did he owe fter 4 moths of o pymet? Aswer Step : Write recursive defiitio Iitil coditio: 0 300( use 0 so tht represets R formul :.08 Step : Use clcultor(press MODE key) the blce fter ecursive 9 moth. After 4 moths, Pierre ows 44.36
Tke ote: A rithmetic sequece is sequece where the differece betwee y two cosecutives terms is costt. This differece is clled commo differece. A recursive defiitio for this sequece hs two prts: d, for iitil coditio recursive formul A explicit defiitio for this sequece is sigle formul: d for, Exmples of Arithmetic sequeces: 3,6,9,,5, Yes it is A.S. d the differece is 3,4,9,6,5, No, there is ot commo differece
Problem 4: Alyzig Arithmetic Sequeces. A. Wht is the 00 th term of the rithmetic sequece tht begis 6,,.? 6,d the commo differece is- 6 5 d 00 6 00 00 50 5 Use the explicit formul Substitute d simplify The 00 th term is 50 B. Wht re the secod d third terms of the rithmetic sequece 00,,, 8,.? 00 d the 4 8. There re 3 commo differeces betwee 00 d 8 8=00+3d -8=3d -6=d The commo differece is -6 d the terms re 00,94,88,8,
Tke ote: the rithmetic me, or verge of two x y Numbers x d y is. The rithmetic sequece, the middle term of y three cosecutives terms is the rithmetic me of the other two terms. Problem 5: Usig Arithmetic Me Wht is the missig term of the rithmetic sequece.,5,, 59,? Aswer: 5 59 37 the missig term is37
roblem 6: Usig Explicit Formul for Arithmetic Sequece The umbers of sets i the first 3 rows i sectio of re form rithmetic sequece. Rows d re show i the digrm below. How my sets re i row 3? Row : 6 sets Step : Fid the commo differece d 6 4 Step : Write explicit formul for 3 d 4 3 38 Row : 4 sets
Wht is series? Tke ote: A SERIES is idicted sum of the terms of sequece. Fiite Sequece: 6, 9,, 5, 8 Fiite Series: 6+ 9+ + 5+ 8 (The sum is 60) Ifiite Sequece: 3, 7,, 5,. Ifiite Series: 3 + 7 + + 5 + A rithmetic series is series whose terms form rithmetic sequece. Whe series hs fiite umber of terms, you c use formul ivolvig the first d lst term to evlute the sum.
ARITHMETIC SERIES S The sum of terms of rithmetic series
Problem : Fidig the sum of Fiite Arithmetic Series Wht is the sum of the eve itegers from to 00? Aswer The series + 4 + 6 + 8 + + 00 is rithmetic with the first term, lst 00 d commo differece d the sum is S Your tur: S 50 50 00 550 Wht is the sum of the fiite rithmetic series 4 + 9 + 4 + 9. +99? Aswer 00 4 d 5 d ( )d 4 ( )5 99 0 5 5 ( )d 00 ( ) 00 50 S 0 0 S 4 99 030
Problem : Usig the sum of fiite rithmetic series A compy pys $0,000 bous to slespeople t the ed of the first 50 wks. if they mke 0 sles i their first week, d the improve their sles umbers by two ech week therefter. Oe slesperso qulified for the bous with the miimum possible umber of sles. How my sles did the slesperso mke i the week 50? I ll 50 wks.? Aswer 0 ( 0 d ) d 50-0 98 08 Use the explicit formul to fid the sles i wk. 50 S 50 0 08 58 950 The slesperso mde 08 sles wk. 50 d 950 sles i ll 50 wks.
5 You c use the Greek cpitl letter sigm to idicte sum. You use limits to idicte the lest d gretest vlue of i the series. For exmple you c write the series s 4 6 8 0 4 6 8 0 Red the sum of s goes from to 5 Tke ote: To write series i summtio ottio you eed explicit formul for the th term d the lower d upper limits. Problem 3: Writig series i Summtio Nottio. Wht is summtio ottio for the series 7 + + 5 +... + 03 + 07? Aswer 7 d 4 d 4 4 3 7 07 5 4 4 3 3 5 = (4 + 3)
Problem 4: Fidig the sum of series Wht is the sum of the series writte i summtio ottio. ) 70 5 3 Aswer Fid 5 3 8 70 Fid 70 570 3 353 Sice the formul is fuctio the series lier is rithmetic S S 70 70 8 353, 635 b) 7 There re 70 terms, so 70 3 4 5 6 7 0 4 9 6 5 36 9
Your tur Aswers 40 00 3 3 8 b) c) - ) 0 ) 40 b) 00 c) 4 Problem 5: Usig G.C. Wht is the sum of the series writte i summtio 70 ottio Aswer 5 3 From the LIST meu. ( d STAT) Your tur 50 Aswer: 4,650
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