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ex_sol(3) - Section 11.3 Geometric Sequences Instructor Ms...

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Unformatted text preview: Section 11.3: Geometric Sequences Instructor: Ms. Hoa Nguyen (nguyenescs.fsu.edu) 1 Geometric Sequences A geometric sequence {an} satisfies the following: . (1,1 . an = art—17' Where a1 is the first term, 1" 7E 0 is the common ratio. SO, 2. Find the n-th term of a geometric sequence 01 (12 = 0417‘ as = a2?" = alrz a," = an_1r = awn—1 So, the n—th term of a geometric sequence is: [an = alrn‘l. | Example 1: [11.3.1aP'fIThe 6“ term of a geometric sequence with first term :11, = —4 and common ratio r = a: is \ n —| 0 ml 6 -I 2 0i : 0\. I" _..1. Q 1 o a n 5 e m: (A) (i _ , J. 2 - 8 Example 2: [11.3.1bP’i‘]Then"‘ term. of a, geometric sequence with first term a] = 6 and common ratio r 2: w2 is o __,(—12)n-1 , 0 ‘None ofthese a”! : 1,31 r‘ - Z a 6(—2)““1 :1; -| o -12»! 2 an: al F‘” 0 “mar” 3‘“-' o —-6(2)"~1 2 02 (- 2) Example 3: [11.3.2aPT]If a geometric sequence has a“; --= ‘13 and a.” a 12, what is thegcomsénon. ratio? CL‘H Que . r» 12 => Y“: Q. 13' $1 46 _ None of these 1: 12 @960 3 Summing the terms of geometric sequence Let 8,609 = '1, ..., n) be the sum of the first 1:: terms of a geometric sequence {an}. Then 81 = an S; = a1+a2=a1+(a1r)=a1(1+r) ,. S3 = a1 + a2 + a3 = a1+(a1r)+(a1r2) = a1(1 +r+ r2) 3,, = _ a1 + a2_+ a3 + + an =, a1 +(a1r)+(a17‘2) +1.. +[017‘n—1] 1— n = a1(1+r+r2+...+r"‘1)=a1 T 1 — 'r if r 7E 1. So, the sum of the first n terms of a geometric sequence is Sn = a1 111’: if r 7E 1. 4 Geometric series A geometric series is a series of the form: 00 z» 00 a1+a1r+a1r2+... = E alrk=a1 16:0 16:0 In other words, the geometric series is a sum of infinite number of terms of a geometric sequence. If |r| < 1, then .900 : alx+ a1?" + alr2 + = infi- Example 4: at 04 u u [11.3.3bP’I‘]Find the sum of the infinite geometric series 1 + § + 3-3- + ‘."+(§)n-—l,+,,, a 0 :14, a : l i" ; j = £ ‘7 i =“’ e 6 ‘ - — r -».«:ql 3 ' G 5 w, km 1: y i 00 ‘ i A__ 4_ Z | Example 5 -— ._ 3 3 Alan 0,,— 11.3.3cPT1Find the sum of the alternating infinite geometric series er, dz \ ‘ a1: '. “all: all" .2) _-——%—-:—.\.“_Ll'_ 0‘41 0 5% _ 4 ‘ ‘ 3 4 8 i 6 l1? a J'— 1 : ,1. l = l _ 2 Li 0 § "L - r" l“ L} é I. 3‘ g 9- - \ 5 Write Repeating Decimal as a Fraction ,6 Example. Repeating Decimal Write 0.898989 « -- as a fraction. Solution. 0.898989 ~ - - = $693 + + + . .. =§§5u+fifi+~f§g+mi =¥&r:§g =3—3— So,a1=%’5,r=fi,5m=g. Example 6: [Il.3.3tiPT]If the repeating decimal 0.135135135- ' - is written as {if in Where m and. n age integers, then m = a $15 0. :55 135+.2»;122,5 s. 5 ‘ a1: E5. r»: ) goo: mtg : 5 e 139 ‘ :9 , [000 ’ " tmg 3 3 3 37 k 9“ Example 7: => W1 3 [11.3.3ePTllf the repeating decimal 0.444444 - » i is written as in re- ‘duced form where. m and n are integers, then m = 0‘ 44 e 13- Oufili-LHHLQ? 5 =‘\_ _ m _ Lt . 4 4&1 7 l~' _3,‘ gou— T‘- _. E. =7 m= Li ...
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