6.09 Practice

6.09 Practice

Geometric Sequence

Need!!!!

• Notes

• Paper & pencil

• Calculator

Sequences

• Arithmetic Sequence: a list of numbers that are adding (subtracting) by the same number each time

• Geometric Sequence: a list of numbers that are multiplied (divided) by the same number each time

Geometric Sequence

Explicit Formula

(write down formula)

a_n = nth term

a₁ = first term

r = common ratio

n = term number

Multiple Choice

What does a₁ represent?

• Any term

• First term

• Last term

Multiple Choice

What does r represent?

• Radical

• Common difference

• Common ratio

Geometric Sequence: 3, 6, 12, 24, 48, 96,...

a_1: 1st term is 3

a₂: 2nd term is 6

a₃: 3rd term is 12

a₄: 4th term is 24

a_5: 5th term is 48

a₆: 6th term is 96

Common Ratio

•  $\frac{2nd\ term}{1st\ term}=\frac{6}{3}=2$  

•  $\frac{3rd\ term}{2nd\ term}=\frac{12}{6}=2$  

•  $\frac{4th\ term}{3rd\ term}=\frac{24}{12}=2$  

•  $\frac{5th\ term}{4th\ term}=\frac{48}{24}=2$  

•  $\frac{6th\ term}{5th\ term}=\frac{96}{48}=2$  

Multiple Choice

Is the sequence -432, 72, -12, 2,... geometric?

• yes

• no

-432, 72, -12, 2, ...

•  $\frac{72}{-432}=-.1666...$  

•  $\frac{-12}{72}=-.1666...$  

•  $\frac{2}{-12}=-\frac{1}{6}=-.1666...$  

• Yes; r = -.1666...

Multiple Choice

Find the common ratio for the geometric sequence.

• 10

• 
  

  $\frac{1}{2}$
  

• 2

• 4

Multiple Choice

_{What is the common ratio (r) for this geometric formula:} a_n = 10(2)^n-1?

• ²

• ¹⁰

• ²⁰

• 2ⁿ

Multiple Choice

Given the sequence: 125, 25, 5, ....

Write the rule using the geometric sequence formula.

• 
  

  $a_n=125\left(\frac{1}{5}\right)^{\left(n-1\right)}$

• $a_n=625\left(\frac{1}{5}\right)^{\left(n-1\right)}$

• $a_n=625\left(5\right)^{\left(n-1\right)}$

• $a_n=125+5n$

Given the sequence: 125, 25, 5, ....

• a₁=125

•  $r=\frac{25}{125}=\frac{5}{25}=\frac{1}{5}=.2$  

•  $a_n=125\left(\frac{1}{5}\right)^{\left(n-1\right)}$  

Example: ____,____,____,____

Find the first four terms of the sequence given the rule: 

 $a_n=3600\left(\frac{1}{2}\right)^{\left(n-1\right)}$  
Start with  $a_1$  ; 3600 and multiply with r;  $\frac{1}{2}$  
 $a_1=3600$ 
 $a_2=3600\cdot\frac{1}{2}=1800$  

 $a_3=1800\cdot\frac{1}{2}=900$  
 $a_4=900\cdot\frac{1}{2}=450$  

Multiple Choice

Find the first four terms of the sequence given the rule:

a_n = 4(5)^n-1

• 20, 100, 500, 2500

• 5, 20, 80, 320

• 4, 20, 100, 500

• 4, 9, 14, 19

a_n = 4(5)^n-1

• Start with 4

• Multiply by 5

• 4, 20, 100, 500

Multiple Choice

What are the next 3 terms in the following geometric sequence:

512, 256, 128, ___, ___, ___, ...

• 1024, 2048, 4096

• 32, 8, 2

• 64, 32, 16

• 92, 66, 38

512, 256, 128, ___, ___, ___, ...

• Find common ratio (r)

•  $r=\frac{256}{512}=\frac{128}{256}=\frac{1}{2}=.5$  

• 4th term: 3rd term x r = 128 x .5 = 64

• 5th term: 4th term x r = 64 x .5 = 32

• 6th term: 5th term x r = 32 x .5 = 16

• 512, 256, 128, 64, 32, 16, ...

Multiple Choice

Write the explicit formula for the geometric sequence.

• a_n = 4(3)^n-1

• a_n = 3(4)^n-1

• a_n = -4(3)^n-1

• a_n = 3(-4)^n-1

Multiple Choice

What is the 9^th term in a geometric sequence with a common ratio of 2 and a first term of 3?

• 384

• 768

• 1,536

• 27

n = 9

r = 2

a₁ = 3

• a₉=3(2)^9-1

• calc: 3(2)⁸

• a₉ = 768

Multiple Choice

Find a₁₂ in the sequence -1, -3, -9, -27, ...

• -177,147

• -19,683

• -59,049

• 118,098

-1, -3, -9, -27, ...

•  $n=12$  
  

   $r=\frac{-3}{-1}=\frac{-9}{-3}=\frac{-27}{-9}=3$ 

•  $a_1=-1$  

•  $a_{12}=-1\left(3\right)^{\left(12-1\right)}$  

•  $a_{12}=-1\left(3\right)^{11}$  

•  $a_{12}=-177,147$  

Multiple Choice

Daisy received $2 on her first birthday. Each year, the amount of money she received tripled. How much money will she receive on her 5^th birthday?

• $10

• $300

• $162

• $416

Daisy received $2 on her first birthday. Each year, the amount of money she received tripled. How much money will she receive on her 5^th birthday?

• a₁=2; r=3; n=5

• a₅=2(3)^5-1

• a₅=2(3)⁴

• a₅=162

• 2, 6, 18, 54, 162