Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Do Now: Aim: What are geometric sequences? Annie deposits $1000 in a bank at 8%. interest is compounded

Aim: Geometric Sequence & Series Course: Alg. 2 & Trig.

Do Now:

Aim: What are geometric sequences?

Annie deposits $1000 in a bank at 8%. interest is compounded quarterly. How much does Annie earn after 1 year?

Balance A, Principal P, rate r, # of compoundings n, time t

A P(1 r

n)nt

Compound Interest

1000(1 + 0.02)4 • 1 = 1082.43


Money in the Bank 1,2,3 Years

Annie deposits $1000 in a local bank at 8%. Interest is compounded annually. How Much does Annie earn after 1 year?Simple interest - paid only on the initial principal

1000 + 1000(0.08) = $1080principal End of year

balanceinterest earned

interest rate

How much does Annie earn after 2 years?

$1000(1.08)(1.08) = 1166.40

How much does Annie earn after 3 years?

$1000(1.08)(1.08) (1.08) = 1259.7121166.40

1080

or 1000(1.08)


Exponential Growth

Principal(1 + interest rate)number of years = Ending balance for 3 years

Post growth y, Pre-growth A

rate r, time t

y = a • bx

y = A(1 + r)t

Exponential function

recall:

In general terms

year 1 2 3

geometric sequence

1166.40 1259.712amount $1080


positive integers 1 2 3 4 · · · n

Definition of Geometric Sequence

A sequence is geometric if the ratios of consecutive terms are the same.Sequence

a1, a2, a3, a4, . . . . . an, . . .

is geometric if there is a number r, r 0, such that

and so on. The number r is the common ratio of the geometric sequence.

,,,3

4

2

3

1

2 ra

ar

a

ar

a

a

1

n

n

ar

a

terms of sequence a1 a1r a1r2 a1r3 a1rn - 1


2, 4, 8, 16, . . . .,

12, 36, 108, 324, . . . . r = 3

r = 2

Definition of Geometric Sequence

A sequence is geometric if the ratios of consecutive terms are the same.Sequence

a1, a2, a3, a4, . . . . . an, . . .

is geometric if there is a number r, r 0, such that

and so on. The number r is the common ratio of the geometric sequence.

,,,3

4

2

3

1

2 ra

ar

a

ar

a

a

r = ?,....81

1,

27

1,

9

1,

3

1 nth? ,....

3

1n

r = -1/3

2n, . . .

4(3)n, . . .

1

n

n

ar

a


The nth term of an geometric sequence has the form

where r is the common ratio between consecutive terms of the sequence. Thus ever geometric sequence can be written in the following form

The nth Term of a Geometric Sequence

a1 a2 a3 a4 . . . . . an . . . .

a1 a1r a1r2 a1r3 . . . .

an = a1rn – 1,

a1rn - 1 . . . . r0

50

40

30

20

10

1 2 3 4

2, 6, 18, 54 r = 3


nth Term of a Geometric Sequence

Find the first five terms of the geometric sequence whose first term is a1 = 3 and whose common ratio is r = 2.

an = a1rn – 1

a1 = a1r1 – 1

a2 = a1r2 – 1

a3 = a1r3 – 1

a4 = a1r4 – 1

a5 = a1r5 – 1

= 3(2)0

= 3(2)1

= 3(2)2

= 3(2)3

= 3(2)4

= 3= 6

= 12

= 24

= 48


Finding a Term

Find the 12th term of the geometric sequence 5, 15, 45, . . .

an = a1rn – 1

a12 = 5(3)11 = 885,735r = 15/5 = 3

Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05.an = a1rn – 1 a15 = 20(1.05)14 39.599r = 1.05

The 4th term of a geo. sequence is 125, and the 10th is 125/64. Find the 14th.

4th term times r6 = 10th term or a10= a4r6

61

64r

2

1r

1024

125

2

1

64

1254

14

a

Consider the 4th term as if it were the 1st term

910 1

61

62 3 41

1 2 3

64

a a r

a r r r r

a a aa r

a a a

a r


Model Problems

Find the seventh term of the geometric progression 32, -16, 8, . . . .

an = a1rn – 1

a7 = 32(-1/2)6 = 1/2r = -16/32 = -1/2


Model Problems

There are three number such that the second is two more than the first and the third is nine times the first. The numbers form a geometric sequence. Find the numbers.

let x = 1st number

x + 2 = 2nd number

9x = 3rd number

2 9

2

x x

x x

,,,3

4

2

3

1

2 ra

ar

a

ar

a

a

1

n

n

ar

a

2 22 9x x

2 2

2

2

9 4 4

8 4 4 0

2 1 0

2 1 1 0

1 / 2, 1

x x x

x x

x x

x x

x x

-½, 1½, -4½

1, 3, 9


Model Problem

A company purchases $50,000 worth of copiers on Jan. 1, 1996. This asset depreciates at a rate of 45% per year. What is the value of this asset at the end of 2001 (or 1/1/02)?

an = a1rn – 1,

(note: a 45% depreciation rate means the asset retains 55% of its value)

a1 = 50,000 r = .55 n = 2002. . . 1996 = 7

a7 = 50000(.55)7-1

a7 1384.032031

Asset is worth approximately $1384.03


Do Now:

Aim: Is it possible to find the sum of a geometric sequence?


Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . .

The Sum of a Finite Geometric Sequence

r = ?3

a1 a2 a3 a4 . . . . . an . . . .

a1 a1r a1r2 a1r3 . . . . a1rn - 1 . . . .

1 3 9 27 . . . . 1(3)7

1 + 3 + 9 + 27 +. . .+ 1(3)7 = 3280

Is there an easier way to find the sum of a geometric sequence?


Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . .

previous problem

The Sum of a Finite Geometric Sequence

The sum of the finite geometric sequence a1, a1r2, a1r3, a1r4, . . . . a1rn - 1 . . . . with common ratio r 1 is given by

1

1

1

n

n

rS a

r

31

311

8

S 32802

6560

r = 3

divergent series – the sum of the series is infinite when |r| > 1


The Sum of a Infinite Geometric Sequence

Find the sum of 1 + 1/2 + 1/4 + 1/8 + . . .

a1 = 1 and r = ?0.5

S =?1

1

1

n

n

rS a

r

sum of the finite

geometric sequence

n = 4 1 + 1/2 + 1/4 + 1/8 = 1 7/8

n = 5 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1 15/16

n = 6 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 1 31/32

n = 3 1 + 1/2 + 1/4 = 1 3/4

as n S approaches 2

1 11

aS r

r


The Sum of a Infinite Geometric Sequence

Find the sum of 3 + 0.3 + 0.03 + 0.003 + . . .

1

1

aS

r

1

1

aS

r

a1 = 3 and r = ?0.1

3 + 3(0.1) + 3(0.1)2 + 3(0.1)3 + . . .

3

1 (0.1)

3

0.9

13

3


Interest Problems

Suppose at the beginning of each quarter you deposit $25 in a savings account that pays an APR of 2% compounded quarterly. Banks post interest at end of quarters. What would be the balance at year’s end?

Date of Deposit

1st year Additions Value at end of

Quarter

Jan 1

April 1

July 1

Oct 1

Account balance at end of year

nt

n

rPA

1

4005.125

3005.125

2005.125

1005.125

25.50

25.38

25.25

25.13

$101.26

The sum represents a finite geometric series where a1 = 25.13, r = 1.005 and n = 4

27.101005.11

005.1113.25

4

S


Model Problems

Find the sum of the series.

160 + 80 + 40 + . . . , n = 6

Find the sum of the infinite series.

160 + 80 + 40 + . . . .


Model Problems

Find the sum of the geometric series.

2 3

3 1, , 5

8 4a a n


Model Problems

Find a1.

Sn = -55, r = -2/3, n = 5


A company purchases $50,000 worth of copiers on Jan. 1, 1996. This asset depreciates at a rate of 45% per year. What is the value of this asset at the end of 2001 (or 1/1/02)?

an = a1rn – 1,

Model Problems


Find the nth term of the geometric sequence.

a2 = -18; a5 = 2/3; n = 6

Model Problems

Documents

Aim: Geometric Sequence & Series Course: Alg. 2 & Trig. Do Now: Aim: What are geometric sequences? Annie deposits $1000 in a bank at 8%. interest is compounded