SECTION 7.3SECTION 7.3

GEOMETRIC SEQUENCESGEOMETRIC SEQUENCES

GEOMETRIC SEQUENCESGEOMETRIC SEQUENCES

(a) (a) 3, 6, 3, 6, 12, 24, 12, 24,

48,48,9696

(b) (b) 12, 4, 12, 4, 4/3, 4/9, 4/3, 4/9,

4/274/27,,

4/814/81

(c) (c) .2, .6, .2, .6, 1.8, 5.4, 1.8, 5.4,

16.2,16.2,

48.648.6

Geometric Sequences have a Geometric Sequences have a “common ratio”.“common ratio”.

(a) r = 2(a) r = 2 (b) r = 1/3 (b) r = 1/3 (c) (c) r = 3r = 3

GEOMETRIC SEQUENCE RECURSION FORMULAGEOMETRIC SEQUENCE RECURSION FORMULA

aa n n = ra = ra n - 1n - 1

This formula relates each term This formula relates each term in the sequence to the in the sequence to the previous term in the sequence.previous term in the sequence.aa n n = 2a = 2a n - 1n - 1

bb n n = 1/3b = 1/3b n - 1n - 1cc n n = 3c = 3c n - 1n - 1

EXAMPLE:EXAMPLE:

Given that aGiven that a 1 1 = 5 and the = 5 and the recursion formula arecursion formula a n n = 1.5a = 1.5a n - n -

11, determine the the value of , determine the the value of aa 5 5 . . aa 2 2 = 1.5(5) = 7.5 = 1.5(5) = 7.5

aa 3 3 = 1.5(7.5) = = 1.5(7.5) = 11.2511.25aa 4 4 = 1.5(11.25) = = 1.5(11.25) = 16.87516.875aa 5 5 = 1.5(16.875) = = 1.5(16.875) = 25.312525.3125

Again, recursion formulas Again, recursion formulas have a big disadvantage!have a big disadvantage!

Explicit Formulas are much Explicit Formulas are much better for finding nth terms.better for finding nth terms.

a a 22 = ra = ra 1 1

a a 33 = ra = ra 2 2 = r(ra = r(ra 1 1 ) = r ) = r2 2 aa 1 1

a a 44 = ra = ra 3 3 = r(r = r(r2 2 aa 1 1 ) = r ) = r3 3 aa 1 1

In general, aIn general, ann = r = rn - 1n - 1aa11

GEOMETRIC SEQUENCE EXPLICIT FORMULA

GEOMETRIC SEQUENCE EXPLICIT FORMULA

PREVIOUS EXAMPLE:PREVIOUS EXAMPLE:

Given that aGiven that a 1 1 = 5 and r = 1.5, = 5 and r = 1.5, determine the the value of adetermine the the value of a 5 5 ..

aa 5 5 = 1.5 = 1.5 4 4 (5) = (5) = 25.312525.3125

EXAMPLE:EXAMPLE:

Given that {aGiven that {ann} = 64, 48, } = 64, 48, 36 . . .36 . . .

determine the value of adetermine the value of a88

First, determine rFirst, determine rr = 48/64 r = 48/64 = .75= .75aa88 = .75 = .7577 (64) (64) aa88

= =

2187

256

EXAMPLE:EXAMPLE:

If a person invests $500 today at If a person invests $500 today at 6% interest compounded 6% interest compounded monthly, how much will the monthly, how much will the investment be worth at the end investment be worth at the end of 10 years (that is, at the end of of 10 years (that is, at the end of 120 months)?120 months)?

The 6% is an annual rate.The 6% is an annual rate.

The corresponding monthly rate The corresponding monthly rate is .06/12 = .005is .06/12 = .005

EXAMPLE:EXAMPLE:

aa11 = 500(1.005) Amt at end of = 500(1.005) Amt at end of mth 1 mth 1

aa22 = 500(1.005) = 500(1.005)22 Amt at end of Amt at end of mth 2mth 2

aa120120 = 500(1.005) = 500(1.005)120120 Amt at end of Amt at end of mth mth 120 120

EXAMPLE:EXAMPLE:

aa120120 = 500(1.005) = 500(1.005)120120 Amt at end of Amt at end of mth mth 120 120

$909.70$909.70

GEOMETRIC SEQUENCE SUM FORMULA

GEOMETRIC SEQUENCE SUM FORMULA

Let aLet a11, a, a22, a, a33 be a geometric be a geometric sequencesequence

Then SThen Snn = a = a11+ a+ a22 + a + a33 + . . . + a + . . . + ann is is the sum of the first n terms of the sum of the first n terms of that sequence.that sequence.

SSnn can also be written as can also be written as

SSnn = a = a1 1 + a+ a11r + ar + a11rr22 + . . . + a + . . . + a11rrn - 1n - 1

GEOMETRIC SEQUENCE SUM FORMULA

GEOMETRIC SEQUENCE SUM FORMULA

SSnn = a = a1 1 + a+ a11r + ar + a11rr22 + . . . + a + . . . + a11rrn - 1n - 1

rSrSnn = a = a11r + ar + a11rr22 + . . . + a + . . . + a11rrn - 1n - 1 + a+ a11rrnn

SSnn - rS - rSnn = a = a11 + 0 + 0 + . . . + 0 + 0 + 0 + . . . + 0 + - a+ - a11rrnn

SSnn (1 - r) = a (1 - r) = a11 (1 - r (1 - rnn))S = a (1 - r

- r or

a ( r - 1r - 1n

1n

1n) )

1

EXAMPLE:EXAMPLE:

Determine the sum of the Determine the sum of the first 20 terms of the first 20 terms of the geometric sequence 36, 12, geometric sequence 36, 12, 4, 4/3, . . . 4, 4/3, . . .

aa11 = 36 = 36 r = 1/3r = 1/3

EXAMPLE:EXAMPLE:

1 - 31

1 - 31

= S20

20

36

53.999999953.9999999

EXAMPLE:EXAMPLE:

If you were offered 1¢ today, If you were offered 1¢ today, 2¢ tomorrow, 4¢ the third day 2¢ tomorrow, 4¢ the third day and so on for 20 days or a and so on for 20 days or a lump sum of $10,000, which lump sum of $10,000, which would you choose?would you choose?

S = .012 - 1

2 - 120

20

= = $10,485.75$10,485.75

This formula is for the sum of This formula is for the sum of the first n terms of a the first n terms of a geometric sequence.geometric sequence.

Can we find the sum of an Can we find the sum of an entire sequence?entire sequence?

For example:For example: 1 + 3 + 9 + 1 + 3 + 9 + 27 + . . .27 + . . .

SUMS OF ENTIRE GEOMETRIC SEQUENCES

SUMS OF ENTIRE GEOMETRIC SEQUENCES

But we can for a sequence But we can for a sequence such assuch as1

2 +

14

+ 18

+ 1

16 + . . .

12

1 - 12

1 - 12

=

n

1 - 1

2

n

11

as nas n

GEOMETRIC SEQUENCE SUM FORMULA

GEOMETRIC SEQUENCE SUM FORMULA

Any geometric sequence with Any geometric sequence with

rr< 1< 1

As nAs n, , S = a (1 - r

1 - ra

1 - rn1

n1)

S = a

1 - r 1

rr< 1< 1

EXAMPLE:EXAMPLE:

Evaluate the sum of the Evaluate the sum of the geometric series:geometric series:

16 + 12 + 9 + 27/4 + . . .16 + 12 + 9 + 27/4 + . . .

r = 3/4r = 3/4 S = 16

1 - 34

6464

EXAMPLE:EXAMPLE:

A ball is dropped from a A ball is dropped from a height of 16 feet. At each height of 16 feet. At each bounce it rises to a height of bounce it rises to a height of three-fourths the previous three-fourths the previous height. How far will it have height. How far will it have traveled (up and down) by traveled (up and down) by the time it comes to rest?the time it comes to rest?

EXAMPLE:EXAMPLE:

Down series:Down series: 16 + 12 + 9 16 + 12 + 9 + . . .+ . . .

Up series:Up series: 12 + 9 + 12 + 9 + 27/4 . . .27/4 . . .S =

16

1 - 34

D S = 12

1 - 34

U

64 + 48 = 112 ft.64 + 48 = 112 ft.

1

3=

k = 1

k

1

3 +

1

3 +

1

3 +

1

3 + . . .

2 3 4

Geometric Geometric SeriesSeries

Recall: a

1 - r1

13

1 - 13

=

1323

= 1

2

EXAMPLEEXAMPLE

3 = k

k = 1

8

3 + 3 3 + 3 22 + 3 + 3 3 3 + . . . + . . . + 3 + 3 88Geometric Geometric

SequenceSequence

Recall: S = a (1 - r )

1 - rn1

n

S = 3(1 - 6561)

1 - 3 = 98408

CONCLUSION OF SECTION 7.3CONCLUSION OF SECTION 7.3