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,...?

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

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

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?

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

• $$S_{\infty}=\frac{a_1}{1-r}$$