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SECTION 7.3 GEOMETRIC SEQUENCES GEOMETRIC SEQUENCES (a) 3, 6, 12, 24, 48, 96 (b) 12, 4, 4/3, 4/9, 4/27, 4/81 (c) .2, .6, 1.8, 5.4, 16.2, 48.6 Geometric Sequences have a “common ratio”. (a) r = 2 (b) r = 1/3 (c) r = 3 GEOMETRIC SEQUENCE RECURSION FORMULA a n = ra n - 1 This formula relates each term in the sequence to the previous term in the sequence. a n = 2a n - 1 b n = 1/3b n - 1 c n = 3c n - 1 EXAMPLE: Given that a 1 = 5 and the recursion formula a n = 1.5a n - 1, determine the the value of a 5 . a 2 = 1.5(5) = 7.5 a 3 = 1.5(7.5) = 11.25 a 4 = 1.5(11.25) = 16.875 a 5 = 1.5(16.875) = 25.3125 Again, recursion formulas have a big disadvantage! Explicit Formulas are much better for finding nth terms. GEOMETRIC SEQUENCE EXPLICIT FORMULA a 2 = ra 1 a 3 = ra 2 = r(ra 1 ) = r2 a 1 a 4 = ra 3 = r(r2 a 1 ) = r3 a 1 In general, an = rn - 1a1 PREVIOUS EXAMPLE: Given that a 1 = 5 and r = 1.5, determine the the value of a 5 . a 5 = 1.5 4 (5) = 25.3125 EXAMPLE: Given that {an} = 64, 48, 36 . . . determine the value of a8 First, determine r r = 48/64 = .75 a8 = .757 (64) a8 = 2187 256 EXAMPLE: If a person invests $500 today at 6% interest compounded monthly, how much will the investment be worth at the end of 10 years (that is, at the end of 120 months)? The 6% is an annual rate. The corresponding monthly rate is .06/12 = .005 EXAMPLE: a1 = 500(1.005) Amt at end of mth 1 a2 = 500(1.005)2 Amt at end of mth 2 a120 = 500(1.005)120 Amt at end of mth 120 EXAMPLE: a120 = 500(1.005)120 Amt at end of mth 120 $909.70 GEOMETRIC SEQUENCE SUM FORMULA Let a1, a2, a3 be a geometric sequence Then Sn = a1+ a2 + a3 + . . . + an is the sum of the first n terms of that sequence. Sn can also be written as S n = a 1 + a 1 r + a 1 r2 + . . . + a 1 rn - 1 GEOMETRIC SEQUENCE SUM FORMULA S n = a 1 + a 1 r + a 1 r2 + . . . + a 1 rn - 1 rSn = a 1 r + a 1 r2 + . . . + a 1 rn - 1 + a 1 rn Sn - rSn = a1 + 0 + 0 + . . . + 0 + - a 1 rn Sn (1 - r) = a1 (1 - rn) n Sn = a1 (1 - r ) 1- r or a1 ( r n - 1) r - 1 EXAMPLE: Determine the sum of the first 20 terms of the geometric sequence 36, 12, 4, 4/3, . . . a1 = 36 r = 1/3 EXAMPLE: 1 36 - 1 3 = 1 -1 3 20 S20 53.9999999 EXAMPLE: If you were offered 1¢ today, 2¢ tomorrow, 4¢ the third day and so on for 20 days or a lump sum of $10,000, which would you choose? S20 = .01 2 20 - 1 2 - 1 = $10,485.75 This formula is for the sum of the first n terms of a geometric sequence. Can we find the sum of an entire sequence? For example: 1 + 3 + 9 + 27 + . . . SUMS OF ENTIRE GEOMETRIC SEQUENCES But we can for a sequence such as 1 2 + n 1 1 1 - 2 2 1 - 1 2 1 4 = + 1 8 + 1 16 1 1 - 2 + . . . n 1 as n GEOMETRIC SEQUENCE SUM FORMULA Any geometric sequence with r< 1 As n, S = n Sn = a1 1 - r a1(1 - r ) 1 - r r< 1 a1 1 - r EXAMPLE: Evaluate the sum of the geometric series: 16 + 12 + 9 + 27/4 + . . . r = 3/4 S = 16 1 - 64 3 4 EXAMPLE: A ball is dropped from a height of 16 feet. At each bounce it rises to a height of three-fourths the previous height. How far will it have traveled (up and down) by the time it comes to rest? EXAMPLE: Down series: 16 + 12 + 9 + . . . Up series: SD = 12 + 9 + 27/4 . . . 16 1 - 3 4 SU = 12 1 - 3 4 64 + 48 = 112 ft. k 1 3 = k =1 1 + 3 3 1 2 1 + 3 3 1 + 3 4 + . . . Geometric Series Recall: a1 1 - r 1 1 1 3 = = 1 2 2 1 3 3 3 EXAMPLE 8 3 = k 3 + 3 2 + 33 + . . . + 3 8 Geometric Sequence k = 1 n Recall: Sn = S8 = a1 (1 - r ) 1 - r 3(1 - 6561) 1 - 3 = 9840 CONCLUSION OF SECTION 7.3