Geometric Sequences. Geometric Sequence. Geometric sequences have a common ratio.

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1 s A geometric sequece is sequece such tht ech successive term is obtied from the previous term by multiplyig by fixed umber clled commo rtio. Exmples, 6, 8,, 6,..., 0, 0, 0, 80,... Geometric sequeces hve commo rtio. We multiply ech term by tht rtio to get the ext term i the sequece. The recursive formul for geometric sequece is writte i the form = = r where r is the commo rtio is the th term of the sequece is the first term i the sequece r is the commo rtio for the geometric sequece is the vrible i sequece 3

2 = = r = r 3 3 = r = r The recursive formul c become difficult to work with if we wt to fid the 0 th term. Usig the recursive formul, we would hve to kow the first 9 terms i order to fid the 0 th. This souds like lot of work. There is esier wy = = 0 r = = = r r r ( ) ( ( )) = = = 3 r r r r 3 = r 3 = r r r = r ( ( ( ( )))) = r = r r r r = r Rther th write recursive formul, we c write explicit formul. The explicit formul is lso sometimes clled the closed form. To write the explicit or closed form of rithmetic sequece, we write = r 6

3 Exmple Give the sequece, 6, 8,, 6,... fid the 0 th term To fid the 0 th term of y sequece, we eed explicit formul for the sequece We see the rtio is 3 Why? Becuse Ad = = 3 7 Exmple Give the sequece, 6, 8,, 6,... fid the 0 th term Our formul is the ( ) = 3 The 0 th elemet is the 0 = 3 ( ) 0 = 3 ( ) 9 = 39,366 8 Exmple ( ) = 3 = ( ) = 3 = ( ) 3 3 = 3 = 3 ( ) = 3 = 0 ( ) = 3 = commo rtio is 3 Fid the first five terms d the commo rtio of the geometric sequece = 3 ( ) 9 3

4 Prctice Write formul for the th term of the geometric sequece 7, 8,, 8,... Do ot use recursive formul We see = = 8 = 8 = Our closed formul is the ( ) = 7 0 Exmple Fid the explicit formul for sequece where r = d =,336 We kow tht ( ) = = 336 ( ) = = = 7 Our explicit formul ( ) = 7 Geometric Series We c fid the sum of fiite umber of terms of geometric sequece S = r i We c lso fid the sum of ifiite geometric sequece if r < S = r i

5 Prtil Sum We wt formul for the fiite sum of geometric series We write the sum S = + r + r + + r + r The multiply by r Subtrct rs -S rs = r + rr + rr + + rr + rr rs = r + r + + r + r rs S = r S ( ) ( r = r ) r S = r S i = r Solve for S 3 Fid the sum of the 0 Exmple fiite geometric series. ( ) i We use the formul for the sum S 0 0. = = 0. = r S = r Sum of Ifiite Geometric Series I the specil cse tht r <, the ifiite sum exists d hs the followig vlue: S = r S = r i

6 Exmple Fid the sum of the ifiite series = i = 3 We use the formul for the sum S = r S = = = Exmple Write to exct frctio = = = = + 00 = which is geometric series S = + = = 90 7 Exmple Write s exct frctio Esier method Multiply by 0000 Multiply by 00 Subtrct Solve for S S = S = S = S = 8 8 S = =

7 Prctice Write s exct frctio 0S = S = S = 9 S = 9 Exmple A hs two job offers () $3,000 to strt with ul icreses of 6% for the first yers () $37,000 to strt with ul icreses of 3% for the first yers Which job should she tke? 0 $3,000 to strt with ul icreses of 6% for the first yers 3,000 (.06).06 = 3,000 = $97, $37,000 to strt with ul icreses of 3% for the first yers 37,000(.03).03 = 37,000 = $96,

8 Exmple A hdsome d very rich old m offers prospective bride (if she will wed him) $000 o their first dy together d 9/0 of the previous dy o followig dys. Fid the most the bride could receive? 9, = = $0,