THE GEOMETRIC SERIES

Transcription

1 Mthemtics Revisio Guides The Geometic eies Pge of M.K. HOME TUITION Mthemtics Revisio Guides Level: A / A Level AQA : C Edexcel: C OCR: C OCR MEI: C THE GEOMETRIC ERIE Vesio :. Dte: Exmples 7 d e copyighted to thei espective owes d used with thei pemissio.

2 Mthemtics Revisio Guides The Geometic eies Pge of The Geometic eies. (lso kow s the Geometic Pogessio o G.P.) I G.P. successive tems e liked by commo tio, such s i, 6,,, 8... The fist tem is deoted by d the commo tio is. A ecusive defiitio of G.P. is theefoe u ; u k+ u k ; tem is obtied fom the pevious oe by multiplyig by the commo tio. The exmple, 6,,, 8... is thus defied s u ; u k+ u k. The tems of G.P. tke the fom,,,,... d the th tem is -. ummig Geometic eies. A geometic seies is fomed by ddig togethe the tems of geometic pogessio. Its sum is give by Multiplyig thoughout by we obti the esult ubtctig the secod esult fom the fist gives - - Both sides c the be fctoised: ( ) ( - ) This gives the fomul fo the sum of geometic seies: ( - ) Whe >, it is moe coveiet to wite the fomul s ( - ) ( ) If <, the teds to zeo with icesig, d theefoe ( - ) teds to. As teds to ifiity, the sum of the geometic seies coveges to the vlue This is kow s the sum to ifiity of the geometic seies.

3 Mthemtics Revisio Guides The Geometic eies Pge of Exmple(): Fid the sum of G.P. with 7 tems, whose thid tem is d commo tio is ½. We must fid the missig fist tem befoe substitutig ito the fomul. u. ice ½, 76. ice <, we c the ete the vlues ito the geel fomul ( ) to obti Exmple(): A G.P. hs 6 positive tems, the secod beig 96 d the fouth beig 6. Fid its sum. We hve u 6 d u 96. Hece 6 9, d thus (positive solutio) d We will use the secod fom of the fomul, ( ) to compute the sum ( ) 6 0. Exmple (): Fid the sum of G.P. whose fist tem is, whose lst tem is 878, d whose commo tio is. Hee, u d u This theefoe gives Fom this, l l The G.P. theefoe hs 7 tems d its sum c be woked out s Exmple(): Fid the sum to ifiity of the G.P. i Exmple (). The fist tem is 76 d the commo tio is ½. ice <, thee is sum to ifiity, we c the substitute the vlues ito the geel fomul to obti the fil esult 76.

4 Mthemtics Revisio Guides The Geometic eies Pge of Exmple (): Fid the commo tio of G.P. with fist tem of d sum to ifiity of. Rege s d substitute to give d filly. Exmple (6): The fist thee tems of geometic seies e k +, k d k 6 whee k is positive itege. Fid the vlue of k d the sum to ifiity of the seies, givig the esult i exct fom. ice we hve G.P., k k 6. Coss-multiplyig, (k +) (k-6) k. k k k 7k -0 k k 7k k 0) (k + ) 0. Hece (positive) k 0 d the fist thee tems of the G.P. e, 0 d. 0 The commo tio,. The sum to ifiity is thus. Agi, vit questios o G.P s sometimes come up i exms. Exmple (7): A geometic seies hs fist tem d commo tio. Fid the lest umbe of tems the seies c hve if its sum exceeds 000. (Copyight OUP, Udestdig Pue Mthemtics, dle & Thoig, IBN , Execise 8B, Q.6) We eed to fid vlue of such tht ( ) > 000. ubstitutio gives ( ) > 000 > 800 > 80 l80 l Tkig logs of both sides, l l 80, The ext itege is 7, so the G.P. eeds to hve 7 tems fo its sum to exceed 000.

5 Mthemtics Revisio Guides The Geometic eies Pge of Exmple (8): A geometic seies hs fist tem 0 d commo tio of. Fid the lest umbe of tems the seies c hve fo its sum to exceed 900. ( ) We eed to fid vlue of such tht > 900. ubstitutio gives 0 ( 6 > 900 ) Tkig logs of ech side, l l 6 6, d filly 960 (multiply by d thus evese sig) l l The ext itege is 0, so the G.P. eeds to hve 0 tems fo its sum to exceed 900. N.B. Note the evesl of the iequlity sig i the lst step. A umbe less th hs egtive logithm, so tht step ivolves dividig by egtive qutity. Exmple (9): The sum of the fist two tems of G.P. is 6, d its sum to ifiity is 00. Give tht the commo tio is positive, fid the tio d the fist tem. We hve hee 00, d ( ) 6. Fom the expessio fo, we hve 00( ) d fom the expessio fo, ( ) 6. olvig simulteously, substitutig 00( ) ito the equtio ( ) 6 gives 00( )( ) 6, o 00(- ) 0.6, d hece commo tio of To fid the fist tem, we ege the fomul fo to give (-), givig 00( 0.8) o 0. The fist tem of the G.P. is theefoe 0 d the commo tio is 0.8.

6 Mthemtics Revisio Guides The Geometic eies Pge 6 of Exmple (0): The film idusty uses ccepted fomul to estimte totl box-office tkigs, ssumig 0% week-o-week declie fte the fist week. The film The Golde Rig Choicles cost millio to poduce, d ws oigilly estimted to tke millio i its fist week. i) Usig the 0% declie fomul, fte how my weeks ws the film expected to bek eve, with the box-office tkigs pyig bck the poductio costs? ii) The ctul tkigs fo the film wee 9millio i the fist week d 6 millio i the secod. how tht the film would be uble to py fo itself if subsequet box-office tkigs followed the sme ted. i) We hve G.P. whose fist tem is d whose commo tio is (00-0)%, o 0.6, o. Fist, we fid vlue of such tht ubstitutig, > ( ) ( ) >. 0 Tkig logs of ech side, l l (multiply by d thus evese sig), d filly l l.69.. (Iequlity sig evesed the lst step ivolved dividig by egtive umbe). The ext itege is, so the G.P. eeds to hve tems fo its sum to exceed. The film is expected to bek eve t the ed of the fifth week (i.e. its box-office totl is expected to exceed millio). ii) We ow hve G.P. whose fist tem is 9, d whose secod tem is 6. The commo tio is theefoe 6 9 o. We must ty d show tht the sum to ifiity of this seies is less th. ubstitutig ito the geel fomul to obti the fil esult 9 7. The film cot tke moe th 7millio, which is still shot of the poductio cost of millio.

7 Mthemtics Revisio Guides The Geometic eies Pge 7 of Exmple (): Tom took out pesio pl t his wokplce wheeby he would sve 00 pe moth i the fist ye. I subsequet yes, he would icese his mothly cotibutios by 6.% of his pevious ye s cotibutio. i) tte Tom s ul pesio cotibutio fo the fist thee yes, d the commo tio of the geometic sequece fomed by his pesio cotibutios. ii) How much moey did Tom py i the fifteeth ye of his pesio pl? iii) how tht Tom s totl pesio cotibutios ove fiftee yes wee lmost exctly iv) Tom s collegue teve took out ltetive pesio pl whee he pid i pe moth i the fist ye, d ech ye lte he pid i x pe moth moe th he did i the pevious ye. (teve s mothly pymets wee oly evised oce pe ye, so he pid the sme mout ove moths.) Afte yes, teve d Tom hd both pid the sme mout ito thei pesio fuds. How much moey did teve put ito his pesio fud i the th ye of his pl, d wht ws the icese i the mothly pymet, x, to the eest pey? i) Tom pid i 00 pe moth, o totl of i the fist ye. I the secod ye, he pid i %, o 78. I the thid ye, he similly pid i % o Addig 6.% to qutity is the sme s multiplyig it by.06, d so the ul cotibutios fom G.P. with commo tio,, of.06. ii) The th tem of the G.P. with fist tem 00 d.06 is , theefoe Tom pid i i totl i the th ye of his pesio pl. iii) We will use ( ) to compute Tom s totl pesio cotibutios fo the yes. 00 (.78) , iv) teve put i 00 pe um i ye d i totl (the sme mout s Tom). teve s ul cotibutios fom A.P. whose sum to tems,, is d whose fist tem,, is 00. ubstitutio ito the geel fomul ½ ( +(-) d) gives 7.(000 + d) d d d 6.08 I the th ye, teve pid i 00 + d (fom bove) o ( ) 69.. His ul cotibutios icesed by 6.08 ech ye, o.7 ech moth, to the eest pey.

8 Mthemtics Revisio Guides The Geometic eies Pge 8 of Exmple (): The th, 7 th d 6 th tems of A.P. themselves fom G.P. The fist six tems of the A.P. hve sum of. Wht is the commo diffeece of the A.P. d the commo tio of the G.P? (Copyight OUP, Udestdig Pue Mthemtics, dle & Thoig, IBN , Execise 8B, Q.8) We c ete 6 d 6 ito the geel fomul to give ( +d) Fo the th, 7 th d 6 th tems to be i G.P, we c epeset them s b, b d b. Theefoe + d b () + 6d b () + d b () ubtctig () fom () gives 9d b(-) () ubtctig () fom () gives d b(-). () Dividig () by () gives. The commo tio betwee the th, 7 th d 6 th tems is theefoe. Fom this fct we c estblish the vlue of the commo diffeece, d. + 6d ( + d) (6) + d 9( + d) (7) ubtctig (6) fom (7) gives 9d 6( + d) 9d 6 + 8d) 6 + 9d 0. The fil step is to subtct the lst fomul fom the oe estblished t the begiig. 6 +d 6 + 9d 0 Elimitig we hve 6d d d. The commo diffeece of the A.P. is thus.

9 Mthemtics Revisio Guides The Geometic eies Pge 9 of Aithmetic d Geometic Me (Not ll syllbuses). The ithmetic me of two umbes is equl to hlf thei sum. Fo exmple, the ithmetic me of 0 d 0 is ½(0 + 0) o. The umbes 0, d 0 e i ithmetic pogessio with commo diffeece of, o hlf the diffeece betwee the two oigil umbes. The geometic me of two (positive) umbes is equl to the (positive) sque oot of thei poduct. Fo exmple, the geometic me of 0 d 0 is (0 0) o 0. The umbes 0, 0 d 0 e i geometic pogessio with commo tio of, the sque oot of the tio of the secod umbe to the fist. Exmple (). Fid the ithmetic me of ) 8 d ; b) d 7; c) d 9. Give the commo diffeece of the esultig thee-umbe pogessios. The ithmetic me of 8 d is ½(8 + ) o 6, d the commo diffeece is ½( - 8) o 8. The ithmetic me of - d 7 is ½(- + 7) o, d the commo diffeece is ½(7 (-)) o 6. The ithmetic me of d 9 is ½( + 9) o, d the commo diffeece is ½(9 - ) o. Exmple (). Fid the geometic me of ) 6 d ; b) d 7; c) 0. d.6. Give the commo tio of the esultig thee-umbe pogessios. The geometic me of 6 d is (6 ) 8, d the commo tio is ( 6) o. The geometic me of d 7 is ( 7) 6 7, d the commo tio is (7 ) o. The geometic me of 0. d.6 is (0..6) , d the commo tio is (.6 0.) o.

10 Mthemtics Revisio Guides The Geometic eies Pge 0 of Moe o igm Nottio. This method of defiig seies hs ledy bee ecouteed i the sectio o ithmetic seies. The Geek lette sigm () stds fo sum. The seies c be expessed s ( ) The below the symbol is the sttig vlue fo the cout vible,, d the bove it is the edig vlue fo. The ( - ) to the ight of the symbol is the ctul tem itself, expessed s fuctio of. Whe, ( - ) ; whe, ( - ) 6, d so o util the lst tem, 8, coespodig to. Aothe eqully vlid defiitio fo the sme seies is The bove the symbol is ot the sme s umbe of tems i the sum hee. Thee e still tems, but this time the cout vible goes fom 0 to, ot to, d the fomul fo the tem hs bee coespodigly modified. 0 ( ). Geometic seies with egtive commo tio c be little moe wkwd to defie: c be expessed s 6 ( ) ( 7 ) o 0 ( ) ( 6 ) Exmple (): Rewite the followig seies i sigm ottio: () (b) (c) (I ech cse, we will begi with ). () This is G.P. with commo tio 0. d fist tem 6, with tems. The geel tem is u 6(0.) - o (0.). Its sigm ottio equivlet is (0.). (b) This is G.P. with commo tio d fist tem -, with 7 tems. The geel tem c be defied eithe s u -(-) - o s u (-) -. The secod method shows the oscilltig tue of the seies moe clely. Its sigm ottio equivlet is 7 ( ). (c) This is G.P. with commo tio 0. d fist tem, with ifiitely my tems. The geel tem is u (0.) - o (0.). Its sigm ottio equivlet is (0.).

11 Mthemtics Revisio Guides The Geometic eies Pge of Exmple (6): Wite dow the seies (do ot sum them) coespodig to the followig sigm ottios: () 7( ) (b) ( ) (0. ) () is G.P. with tems with commo tio, d whose fist tem is (ot 7!), i.e (b) is G.P., this time with sum to ifiity. Its fist tem is (-) 0 (0.) 0 o, the secod oe is (-) (0.) o 0., the thid (-) (0.) o 0.0, givig commo tio of 0.. It theefoe goes