Geometric Sequences. Definition: A geometric sequence is a sequence of the form

Transcription

1 Geometic equeces Aothe simple wy of geetig sequece is to stt with umbe d epetedly multiply it by fixed ozeo costt. This type of sequece is clled geometic sequece. Defiitio: A geometic sequece is sequece of the fom 4,,,,,... The umbe is the fist tem, d is the commo tio of the sequece. The th tem of geometic sequece is give by. The umbe is clled the commo tio becuse y two cosecutive tems of the sequece diffe by multiple of, d it is foud by dividig y tem + fte the fist by the pecedig tem. Tht is +. Is the equece Geometic? Exmple : Detemie whethe the sequece is geometic. If it is geometic, fid the commo tio. (), 8,,8,... (b),,, 5, 8,... olutio (): I ode fo sequece to be geometic, the tio of y tem to the oe tht pecedes it should be the sme fo ll tems. If they e ll the sme, the, the commo diffeece, is tht vlue. tep : Fist, clculte the tios betwee ech tem d the oe tht pecedes it

2 Exmple (Cotiued): tep : Now, compe the tios. ice the tio betwee ech tem d the oe tht pecedes it is 4 fo ll the tems, the sequece is geometic, d the commo tio 4. olutio (b): tep : Clculte the tios betwee ech tem d the oe tht pecedes it tep : Compe the tios. ice they e ot ll the sme, the sequece is ot geometic. imil to ithmetic sequece, geometic sequece is detemied completely by the fist tem, d the commo tio. Thus, if we kow the fist two tems of geometic sequece, the we c fid the equtio fo the th tem. Fidig the Tems of Geometic equece: Exmple : Fid the th tem, the fifth tem, d the 00 th tem, of the geometic sequece detemied by 6,. olutio: To fid specific tem of geometic sequece, we use the fomul fo fidig the th tem. tep : The th tem of geometic sequece is give by o, to fid the th tem, substitute the give vlues 6, ito the fomul. 6

3 Exmple (Cotiued): tep : Now, to fid the fifth tem, substitute 5 ito the equtio fo the th tem tep : Filly, fid the 00 th tem i the sme wy s the fifth tem Exmple : Fid the commo tio, the fifth tem d the th tem of the geometic sequece. (), 9, 8, 79,... (b) t t t,,,, olutio (): I ode to fid the th tem, we will fist hve to detemie wht d e. We will the use the fomul fo fidig the th tem of geometic sequece.

4 Exmple (Cotiued): tep : Fist, detemie wht d e. The umbe is lwys the fist tem of the sequece, so. The tio betwee y tem d the oe tht pecedes it should be the sme becuse the sequece is geometic, so we c choose y pi to fid the commo tio. If we choose the fist two tems 9 9. tep : ice we e give the fouth tem, we c multiply it by the commo tio 9 to get the fifth tem. 5 4 ( ) tep : Now, to fid the th tem, substitute, 9 ito the fomul fo the th tem of geometic sequece. ( )( 9) ( 9)

5 Exmple (Cotiued): olutio (b): tep : Clculte d. t 6 t 6 t tep : The fifth tem is the fouth tem multiplied by the commo tio. Theefoe, 5 4 t t 54 4 t 6 tep : Now, substitute t, ito the fomul fo the th tem. t Ptil ums of Geometic equece: We c stt developig fomul fo the sum of the fist tems of geometic sequece,, by witig it out i log fom

6 Next, we multiply both sides by We subtct the fist esult fom the secod ( ) ( ) ( ) ( )... ( ) Usig the commuttive d ssocitive popeties to ege the tems o the ight, so if, 4 4 ( ) + ( ) + ( ) + ( ) ( ) ( ) ( ) ( ). Defiitio: Fo the geometic sequece, the th ptil sum ( ) is give by Witte usig summtio ottio, the th ptil sum of geometic sequece is i k. i This epesets the sum of the fist tems of geometic sequece hvig fist tem k k d commo tio.

7 Exmple 4: Fid the ptil sum coditios. of the geometic sequece tht stisfies the give (),, 7 (b) 5 i ( 8)( ) i olutio (): To fid the th ptil sum of geometic sequece, we use the fomul deived bove. tep : To use the fomul fo the th ptil sum of geometic sequece, we oly eed to substitute the give vlues,, 7 ito the fomul. 7 7 () 8 7 olutio (b): This is the sum of the fist five tems of the geometic sequece with 4. tep : ice the ptil sum is give i summtio ottio, we must fist fid d. Fom the give ifomtio, we kow k 8,, 5. o, k ( 8) 4

8 Exmple 4 (Cotiued): tep : Now tht we kow 4,, we c substitute these vlues ito the fomul fo the th ptil sum to fid the fifth ptil sum. Ifiite eies: A expessio of the fom is clled ifiite seies. The dots me tht we e to cotiue the dditio idefiitely. The ide of ddig ifiitely my umbes d gettig fiite umbe my seem stge, but coside the followig sceio. To begi with, sil is 00 feet fom tee. O the fist dy, it tvels hlf the distce to the tee. O the secod dy, it tvels hlf the emiig distce to the tee, d o the thid dy hlf of the emiig distce gi. This pocess of tvelig hlf the emiig distce pe dy c cotiue idefiitely d t the ed of ech dy some distce will still emi. ee the followig figues.

9 Does this me tht the sil will eve ech the tee? Of couse ot. Let s dd up the distce ou sil hs tveled:

10 This is ifiite seies, d we ote two thigs bout it: Fist, o mtte how my tems of this seies we dd, the totl will eve exceed 00. ecod, the moe tems of this seies we dd, the close the sum is to 00. This suggests tht the umbe 00 c be witte s the sum of ifiitely smlle umbes: To mke this clee, let s look t the ptil sums of this seies: d the geel fom As gets lge d lge, we e ddig moe d moe of the tems of this seies. Ituitively, s gets lge, gets close to the sum of the seies. Now otice tht s gets lge, / gets close d close to zeo. Thus gets close to Usig the ottio of ectio 5.5, we c wite 00 s. I geel, if ifiite seies. gets close to fiite umbe s gets lge, we sy tht is the sum of the Ifiite Geometic eies: A ifiite geometic seies is seies of the fom

11 We leed elie i this sectio tht the sum give by of the fist tems of geometic sequece is ( ). If <, tht is, if < <, the tem stedily deceses i bsolute vlue s iceses, gettig close d close to zeo. The fct tht gets ee d ee to zeo s tkes o lge d lge vlues suggests tht the sums should themselves be gettig close d close to some ceti vlue. This is ctully wht hppes, d we wite s. If > d 0, the tems icese i bsolute vlue without boud s iceses without boud, d the sums do lso. If d 0, does ot exist s. Defiitio: If <, the the sum of the ifiite geometic seies is.

12 Exmple 5: Fid the sum of the ifiite geometic seies olutio: To fid the sum of ifiite geometic seies, we use the fomul tep : Fist, we must fid d. This is doe i the sme wy s fo geometic sequece. The vlue is lwys the fist tem i the seies, d the commo tio is the tio betwee y tem d the oe pecedig it. o, tep : Now substitute the vlues 4, ito the fomul fo the sum of ifiite geometic seies