Geometric Sequences

Geometric Sequences

Two different ways to show the same thing

Some text here about the topic of discussion

​Explicit Formula

​Recursive Formula

Multiple Choice

What would term a₁ be in the sequence:

4, 8, 16, 32, 64, ...

• 16

• 8

• 4

• 32

Multiple Choice

What is the common ratio (r) in the sequence:

4, 8, 16, 32, 64, ...

• 2

• 4

• 6

• 32

Multiple Choice

What is the common ratio (r) in the sequence:

-2, 16, -128, 768, -6144, ...

• 2

• 8

• -8

• 4

Multiple Choice

Using the sequence: 5, 10, 20, 40, 80, 160, ...

If a₄ = 40, what would a_n-1 be?

• 5

• 10

• 20

• 40

Find a₁ ---> the first term

​

Understand what a_n-1 means.... it means the previous term​

​

​Recursive Formula

Multiple Choice

Given the sequence: 3, 12, 48, 96, 384, ...

What is a₁?

• 3

• 12

• 48

• 96

Multiple Choice

Given the sequence: 3, 12, 48, 96, 384, ...

What is r?

• 3

• 4

• 12

• 96

Multiple Choice

Given the sequence: 3, 12, 48, 96, 384, ...

What is the recursive formula?

• a₁ = 3 

  a_n = (4) a_n-1

• a₁ = 4 

  a_n = (3) a_n-1

• a₁ = 12 

  a_n = (9) a_n-1

• a₁ = 3 

  a_n = (-9) a_n-1

a₁ = first term

​

r = common ratio

​

n - 1 = previous term​

​

n = desired term​

​

​

​Explicit Formula

​

Multiple Choice

Given the sequence: 6, 30, 150, 750, 3750, ...

What is a₁?

• 3750

• 5

• 6

• 150

Multiple Choice

Given the sequence: 6, 30, 150, 750, 3750, ...

What is r?

• 3750

• 5

• 6

• 150

Multiple Choice

Given the sequence: 6, 30, 150, 750, 3750, ...

What is the explicit formula?

• a_n = 5*(5)^n-1

• a_n = 30*(6)^n-1

• a_n = 6*(30)^n-1

• a_n = 6*(15)^n-1

Multiple Choice

Given the sequence: 6, 30, 150, 750, 3750, ...

Use the explicit formula from the previous slide to the find the 8th term?

• 30

• 3750

• 475068

• 468750

Example: 120, 60, 30, 15, 7.5, ...

Sequences can also decrease as well. If the common ratio is less than 1, the sequence decays or declines

Common ratio less than 1​

​

64, 32, 16, 8, 4, ...​

​

r = 1/2​

Decay

Common ratio greater than 1

​

4, 8, 16, 32, 64, ...​

​

r = 2​

Growth

The common ratio determines if the sequence is growth vs. decay

Multiple Choice

Is the following sequence a growth or decay?

7, 49, 343, 2401, 16807, ...

• Growth

• Decay

Multiple Choice

Is the following sequence a growth or decay?

-20, -60, -360, -2160, -12,960, ...

• Growth

• Decay

Multiple Choice

Find the next three terms: 4, 12, 36, 108, ...

• 324, 972, 2916

• 972, 2916, 8748

• 405, 1215, 3645

Multiple Choice

What is the recursive rule for the sequence:
11, 22, 44, 88...

• a_n = 2(11)^n-1

• a_n = 11x 2

• a_n = 11(2)^n-1

• a_n = 2a_n-1

Multiple Choice

Given the sequence:
125, 25, 5, ....
Write the explicit rule:

•  t(n) = (125)(5)⁽ⁿ⁾

•  t(n) = 625(1/5)⁽ⁿ⁾

•  t(n) = 625(5)⁽ⁿ⁾

•  t(n) = 125 + 5n

Multiple Choice

Find the general rule for: 64, -32, 16, -8, …  

• 64(1/2)^n-1

• (-32)^n-1

• 64(-1/2)^n-1

• (32)^n-1

Multiple Choice

Which sequence is represented by the equation 2(7)^n-1 ?

• 7, 14, 28, 56...

• 2, 14, 98, 686...

• 2, 9, 16, 23...

• 7,9,11,13...

GEOMETRIC SERIES

The sum of the terms in a geometric sequence.

GEOMETRIC SERIES

GEOMETRIC SERIES

Multiple Choice

What is the formula to be used in finding the sum of the  first five terms of the sequence 5,15,45,...?

• $$S_n=\frac{n}{2}\left(a_1+a_n\right)$$

• $$S_n=\frac{a_1\left(1-r^n\right)}{1-r}$$

Fill in the Blanks

What is the sum of the first five terms of the sequence 5,15,45,...? On the problem, what is the first term?

Fill in the Blanks

What is the sum of the first five terms of the sequence 5,15,45,...? On the problem, what is the common ratio?

Fill in the Blanks

On the problem, how many terms are to be added?

Fill in the Blanks

What is the sum of the first five terms of the sequence 5,15,45,...?

INFINITE GEOMETRIC SERIES

INFINITE GEOMETRIC SERIES

Multiple Choice

Find the indicated sum of the geometric sequence if its first term is 318 and the common ratio is one-half. Is the problem an example of INFINITE GEOMETRIC SERIES?

• Yes, it is!

• No, it is not!

Multiple Choice

What is the formula to be used to solve the problem?

• $$S_n=\frac{a_1\left(1-r^n\right)}{1-r}$$

• $$S_{\infty}=\frac{a_1}{1-r}$$

Fill in the Blanks

Find the indicated sum of the geometric sequence if its first term is 318 and the common ratio is one-half.