Geometric Series

Geometric Series

GOAL: To be able to determine the difference between an arithmetic series and geometric series while applying similar ideas to work with them.

Multiple Choice

Is the sequence-4, -12, -36, -108,... arithmetic geometric, or neither?

• arithmetic
• geometric
• neither

Multiple Choice

What kind of sequence is the pattern 400, 200, 100, 50, 25, ...?

• Arithmetic
• Geometric
• Neither

Multiple Choice

For each sequence, state if it is arithmetic, geometric, or neither.
−34, −26, −18, −10, −2, ... 

• Arithmetic
• Geometric
• Neither

Multiple Choice

Is the sequence-4, -12, -36, -108,... arithmetic geometric, or neither?

• arithmetic
• geometric
• neither

​Definition of a Geometric Series

• ​Recall from yesterday

• Series: The summation of a sequence. (Summation implies the sum of all terms of the sequence.)

• ​Geometric Series

  • ​Summation of a Geometric Sequence

    • There are two types for GEOMETRIC

      • 1. FINITE (meaning the series has a defined last nth term)

      • 2.INFINTE (meaning the series has no defined end)

Multiple Choice

Given the sequence,

4, 12, 36, ..., 78732

Find the value of n given a_n = 78732

• 8

• 9

• 10

• 11

Multiple Choice

Given the sequence,

220, 110, 55, ... , 3.4375

Find the value of n given a_n = 3.4375

• 5

• 6

• 7

• 8

Multiple Choice

Given the sequence,

15, 45, 135, ... , 2657205

Find the value of n given a_n = 2657205

• 9

• 10

• 11

• 12

Multiple Choice

Evaluate the geometric series below: a₁ = -2, a_n = -1458, r = 3

You will need to find the value of n...

• -2186
• -2408
• 1
• -2703

Multiple Choice

Find the sum of geometric series.

−2 + 6 -,..., −162

Remember (-) is simply addition of a negative number.

• -118

• 5

• -122

• 201

Multiple Choice

Evaluate  $$\sum_{n\ =\ 1}^7\left(-2\right)^{n\ -1\ }$$  

• 44

• $$\frac{1}{3}$$  

• 43

• 49

Multiple Choice

The local power company plans to raise rates to cover the increased cost of producing power. They determine the costs will increase by 4.2% per year for the next 10 years. If you paid a total of $1031.60 for power last year, what would you expect to pay for power over the next 10 years? 
(Hint: Find a₁, a₁₀, and r)

• $13,020.97
• $10,316.00
• $2,704.97
• $12,220.97

Multiple Choice

A bouncing ball reaches heights of 16 cm, 12.8 cm, and 10.24 cm on three consecutive bounces. If the ball was originally dropped from 25 cm (a₁ = 25), how much total distance in the downward direction has the ball traveled after five bounces? 

• 25 cm
• 84.04 cm
• 53.7956 cm
• 59.04 cm

Multiple Choice

We say that if r is greater than 1 the series DIVERGES and does not have a sum. If the r value is between 0 and 1 then the geometric series will CONVERGE AND have a sum. Given the geometric series

2 + 4 + 8 + ...

Does the series DIVERGE or CONVERGE?

• Diverges;

  r = 2 which is greater than 1

• Converges; r = 2 which is greater than 1.

• Neither or these, it is not geometric.

Multiple Choice

We say that if r is greater than 1 the series DIVERGES and does not have a sum. If the r value is between 0 and 1 then the geometric series will CONVERGE AND have a sum. Given the geometric series

100 + 50 + 25 + ...

Does the series DIVERGE or CONVERGE?

• Diverges;

  r = 1/2 which is greater than 1

• Converges; r = 1/2 which is less than 1.

• Neither or these, it is not geometric.

Multiple Choice

Given the geometric series;

1000 + 250 + 62.5 ...

Finite or Infinite? Diverges or Converges?

• Finite

• Infinite, DIVERGES; r > 1

• Infinite, CONVERGES; r < 1

Poll

Given the geometric series;

1000 + 250 + 62.5 ...

Which is the correct equation to find the sum?

• $$S_{\infty}=\frac{1000}{1-\frac{1}{4}}$$  

• $$S_{\infty}=\frac{1000}{1-4}$$  

• $$S_{\infty}=\frac{1-\frac{1}{4}}{1000}$$  

Multiple Choice

Given the geometric series;

1000 + 250 + 62.5 ...

What is the sum?

• $$S_{\infty}=1333.3333$$  

• $$S_{\infty}=-333.333$$  

• $$S_{\infty}=.00075$$  

Multiple Choice

Evaluate the infinite geometric series:

1 + 1/5 + 1/25 + ...

• 5/4

• 9/5

• 5/6

• 65/27

Multiple Choice

Evaluate the infinite geometric series:

3 - 2 + 4/3 - 8/9 + ...

• 5/4

• 9/5

• 5/6

• 65/27

Multiple Choice

Find the sum of the infinite geometric series, if it exists. $$\frac{1}{2}-\frac{5}{3}+\frac{50}{9}-\frac{500}{27}+...$$  

• 205

• 19.5

• 108.75

• Does Not Exist