SECTION 7.3 GEOMETRIC SEQUENCES

(a) 3, 6, 12, 24, 48,96 (b) 12, 4, 4/3, 4/9, 4/27, 4/27,4/81 (c).2,.6, 1.8, 5.4, 16.2, 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) = a 4 = 1.5(11.25) = a 5 = 1.5(16.875) =

Again, recursion formulas have a big disadvantage! Explicit Formulas are much better for finding nth terms.

a 2 = ra 1 a 3 = ra 2 = r(ra 1 ) = r 2 a 1 a 4 = ra 3 = r(r 2 a 1 ) = r 3 a 1 In general, a n = r n - 1 a 1 GEOMETRIC SEQUENCE EXPLICIT FORMULA

PREVIOUS EXAMPLE: Given that a 1 = 5 and r = 1.5, determine the the value of a 5. a 5 = (5) =

EXAMPLE: Given that {a n } = 64, 48, determine the value of a 8 First, determine r r = 48/64 =.75 a 8 =.75 7 (64) a8 =a8 =a8 =a8 =

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: a 1 = 500(1.005) Amt at end of mth 1 a 2 = 500(1.005) 2 Amt at end of mth 2   a 120 = 500(1.005) 120 Amt at end of mth 120

EXAMPLE: a 120 = 500(1.005) 120 Amt at end of mth 120 $909.70

GEOMETRIC SEQUENCE SUM FORMULA Let a 1, a 2, a 3 be a geometric sequence Then S n = a 1 + a 2 + a a n is the sum of the first n terms of that sequence. S n can also be written as S n = a 1 + a 1 r + a 1 r a 1 r n - 1

GEOMETRIC SEQUENCE SUM FORMULA S n = a 1 + a 1 r + a 1 r a 1 r n - 1 S n = a 1 + a 1 r + a 1 r a 1 r n - 1 rS n = a 1 r + a 1 r a 1 r n a 1 r n S n - rS n = a a 1 r n S n (1 - r) = a 1 (1 - r n )

EXAMPLE: Determine the sum of the first 20 terms of the geometric sequence 36, 12, 4, 4/3,... a 1 = 36r = 1/3

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? = $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:

SUMS OF ENTIRE GEOMETRIC SEQUENCES But we can for a sequence such as 1 as n 

GEOMETRIC SEQUENCE SUM FORMULA Any geometric sequence with  r  < 1 As n  ,  r  < 1

EXAMPLE: Evaluate the sum of the geometric series: / r = 3/4 64

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: Up series: / = 112 ft = 112 ft.

Geometric Series

EXAMPLE Geometric Sequence

CONCLUSION OF SECTION 7.3 CONCLUSION OF SECTION 7.3