Geometric Sequences

A sequence whose values follow a pattern of multiplying the same constant (not zero)

times each term to arrive at the next. is referred to as a

The constant being multiplied is called the

GeometricExelicit formula:

GeometricRecursive formula

rea

I An explicit formula can easily be used to

find any term in a sequence.

Just plug the number of the term you

want, in for n.

I A recursive formula always has two parts, 1

I the first term, and what you need to do to get 1

I more terms. For a geometric sequence, it

would be what you need to multiply by each 1So, if you want the 8th term, all you need to 1

I do is plug in an 8 for the n in the formula. I time.

I Therefore, it is not as easy to use if you

r is the common ratio

n is the number of the term to find

al is the first term of the sequence (n = 1)

an is the nth term of the sequence

an-I is the term before the nth term

just need to find, say, the 8th term. Because

you would need the 7th term, and to find that

you need the 6th term, and to find that you

need the 5th term, and so on, back to the 1st

term.

GSE Alaebra I

Natne:

Comparing Sequences Homewo

will j/ 'j/ JMario and IUigi both want YOU to corne (Inve 00 ts lot theit

coins. Each one nnakes an otter:

Mario: I will give YOU 3 gold coins on the fust (joy, lhen, ovety doy 't, I Will

much as I paid you the day before.

Luigi: I will give YOU 3 gold coins on the first day. Then, every day after that, I will pay you

than I paid YOU the day before.

Who would YOU rather work fore Use the table below to help you decide.

Mario's Deal Dail Wa e Lui i's DealMonday

Tuesday

Wednesday

Thursday

Friday

Total Earnings

27

Monday

Tuesday

Wednesday

Thursday

Friday

Total Earnings

Daily Wage

-3

2-3

3

l) Whom would you rather work for?

Why?

2) Does the deal represent an arithmetic or geometric sequence? Explain why.Mario

Luigi L(-ntar c

3) Why does Mario's deal ventually become better than Luigi's deal?—ponenha/i% '(watts move nap/d/%

4 Write an ex [cit function tore resent each se uence.Mario's DealLui i's Deal

an: 90h / 7

Geotnetric Sequences Notes

GeometricSequences;Asequence ofterms that havea

Geometric Recursive Formula:

an = r(an-I)

ra+iD between them.

Geometric Explicit Formula:

anta,

Where at is the-hr-4--— in the sequence and r is the common fthD

Exam les

First term Common ratio

1, 2, 4, 8, 16

2 210, —2, —

5' 25 '

5, 15, 45, 135

320, 80, 20, 5, ..

4, 8, 16, 32 ...

256, 64, 16, 4

414 -S/ 4

-4, -12, -36, -108 ...

03

-L

4

2S(p

—4

Recursive Formula Explicit Formula

n-l

6

gn = •gn-l

320320.4 n I

4

25•gn-l gn — —L)n-l

3 •gn-l

Examples:

1. Find the first five terms of the geometric sequence defined as follows:

-l — , -27 -81

2. Find the first five terms of the geometric uence defined as follows:

= 216 gn=;• gn-l

3. Find the common ratio and the missing term in the sequence

7, 21, 63, U 5673

For 4 & 5, Write the recursive and explicit formula for the sequences below.4. 8, 16, 32, 64 t Explicit: gng—• (A)n-l Recursive:

s. 200, 100, 50, 25, ... —L Explicit: (b) n-l Recursive: gl

6. A colony of ants starts with 5 members. The colony triples every year.a. Identify the first term and the common ratio: gl =

b. Write an explicit function

3n-l

¯¯

to represent the sequence. gn-

c. Write the recursive formula to represent the sequence. gl = ; gn=3 gn 1

d. How many members will the colony have after 3 years? 45 5 (3)

e. How many years will it take for the colony to reach greater than 1,000 ants?

abac+5, Los' 121<

Geometric Sequences CW Name

Find the next three terms in each geometric sequence.

1. 2, 4, 8, 16 32

2. 400, 200, 100, 50 as

3. 4,-12, 36, -108, ... -3.24—0

12-8

G,zs29 It,

Find the missing term(s) in each geometric sequence.

4.

Write the recursive rule and explicit rule for each geometric sequence.

6. 9, 27, 81, 243 3 Explicit:

7. Explicit:

8. 12, 3, 3/4, 3/16, . Explicit:

9. 1, 3, 9, 27 3 Explicit:

10. 2, 8, 32, 128 Explicit:

11. 125, 25, 5, 1, ... Explicit:

q • ( )n-l Recursive: gl

( —l ) n-l Recursive: gl =

12 • (A)n-l Recursive: gl

—L • (—3-)n-l Recursive:

Z • Recursive:

12—6 • ( -L ) n- l Recursive:

Find the nth term for each sequence. Round to 3 decimal places if necessary.

12. gl = 3, r = -4, n = 6 Explicit: (1.4-) n-l

13. gl —--500, r = 1/2, n=10 Explicit: gn=-5m• (4) n- 1 gl() =

Write the explicit formula and find the indicated term for each sequence.

14. g6 for 124, 62, 31, 15.5... Explicit: (-4) n

15. for -2, 4, -8, 16... Explicit: g12

Find the next three terms of the sequence.

16. The first term is 6 and the common ratio is -4. Find the next 3 terms.

-214

17. The first term is 2500 and the common ratio is 1/5. Find the next 3 terms.

gn =

4

-z -307:

12

2409b

2.500

Find the indicated term of the sequence.

t8. The first term of a geometric sequence is 1, and the cornmon ratio is 10. What is the 10th term of the

sequence?

19. What is the 11 th term of the geometric sequence 3, 6, 12, 24,

3072-

Applications:

20. A construction company is building houses in a neighborhood. During the 1st month they built 3 homes,

during the 2nd month they built 6 homes, and during the 3rd month they built 12 homes.

a. Write the recursive rule for the sequence. gl _Z_ •gn-l

b. Write the explicit rule for the sequence. 3 • (A n-I

c. the construction company build %homes? S

21. The football team made it to the state playoffs for the 4th year in a row! The chart shows the number

of teams in the playoffs each week. Write the recursive and explicit formula geometric sequence

showing the model:

Round

Teams1 2

32 16

3

8

44

5

2

a. Write the recursive rule for the sequence.

b. Write the explicit rule for the sequence. gn=n (+) n-l

22. It's time to call the exterminator! You found out that the number of termites under your house is

tripling every week. If you have 8 termites on week 1, find the following:

a. A sequence to show the growth of termites.

b. The number of termites after 12 weeks (yikes!!).