Aim: Geometric Sequence Course: Math Literacy Do Now: Aim: What are geometric sequences? Mrs....

Aim: Geometric Sequence Course: Math Literacy

Do Now:

Aim: What are geometric sequences?

Mrs. Gonzales sells houses and makes a commission of $3750 for the first house sold. She will receive a $500 increase in commission for each additional house sold. How many houses must she sell to reach total commissions of $65000?


Model Problem

Mrs. Gonzales sells houses and makes a commission of $3750 for the first house sold. She will receive a $500 increase in commission for each additional house sold. How many houses must she sell to reach total commissions of $65000?

Sn = 65000 a1 = 3750 d = 500

)(2 1 naan

S

an = a1 + (n – 1)d)3750(

265000 na

n

)500)1(3750(3750(2

65000 nn

25007000130000 nn

n = 10.58 and –24.58

She must sell 11 houses.


Geometric Sequence

Determine the next three terms in the following sequences

1, 2, 4, 8, . . .

0.1, 1, 10, 100, . . .

336, 168, 84, 42, . . .

900, 300, 100, . . .


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, . . .


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 . . . .

recursive definition: an = an – 1 · r


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 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


Model Problem

You want to a reduced copy of a photo. The actual length of the photo is 10 in. The smallest size the copier can make is 64% of the original. Find the length of the photo after five reduction at 64%.

an = a1rn – 1

a1 = 10 r = .64 n = 6

a6 = 10(.64)6-1

a6 1.07

After reduction the photo is about 1 in. long.

Aim: Geometric Sequence Course: Math Literacy Do Now: Aim: What are geometric sequences? Mrs....

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