10/31/2023_6th Hour_Arithmetic and Geometric Sequences

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​What is a sequence?

• ​a particular order in which related events, movements, or things follow each other.

​

• a set of related events, movements, or things that follow each other in a particular order.​

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What is an example of a sequence? 

Think of examples that are not numbers.

Arithmetic and Geometric Sequences

​Sequences

Multiple Choice

Arithmetic, geometric, or neither?

4, 9, 14, 19, ...

• Arithmetic

• Geometric

• Neither

• Fibonacci

Multiple Choice

Arithmetic, geometric, or neither?

1, 3, 7, 13, ...

• Arithmetic

• Geometric

• Neither

• Fibonacci

Multiple Choice

Arithmetic, geometric, or neither?

100, 50, 25, 12.5, ...

• Arithmetic

• Geometric

• Neither

• Fibonacci

4, 7, 10, 13, ...

2, 10, 50, 250, ...

​Characteristics

EXPLICIT FORMULAS

• find any term in a sequence ​​

• if you know any 3 variables then you can find the 4th variable.​

Arithmetic sequence

Geometric sequence

100, 20, 4, 4/5, ...

​Writing the explicit formula

0.25, 0.75, 1.25, 1.75 ....

Multiple Select

How well do you understand this topic so far?

• I love it!

• I would like some more help!

• I don't have an opinion

• It's going well

• I don't understand

Multiple Choice

What is the common ratio of the geometric sequence below?
5, 25, 125, 625...

• 25
• 20
• 5
• 4

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

Multiple Choice

Find the common ratio: 64, -16, 4, -1, ...

• r = -8
• r = -1/4
• r = -4
• r = 4

Multiple Choice

Given the sequence:
 25, 21, 17, 13,...
Write the explicit equation that models the sequence.  (Hint:  Simplify your equation.)

• a_n = 4n + 29
• a_n = -4n + 25
• a_n = -4n + 29
• a_n = 4n + 25

Multiple Choice

Find the 10th term.

• 16
• 12
• 21
• 30

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

This image shows how a certain bacteria grows in a petri dish. What is the common ratio of this sequence?

• 4
• 3
• 2
• 6

Multiple Choice

Write the explicit formula for the sequence.

216, 72, 24, 8, ...

• $$a_n=216\cdot\left(\frac{1}{3}\right)^{\left(n-1\right)}$$

• $$a_n=8\cdot3^{\left(n-1\right)}$$

• $$a_n=216-144\left(n-1\right)$$

• $$a_n=216\cdot3^{\left(n-1\right)}$$

Multiple Choice

Which of the following is an example of an arithmetic sequence?

• 4, 8, 16, 32, 64

• 1, 2, 4, 7, 11

• 15, 13, 11, 9, 7

• 3, 6, 12, 24, 48

Multiple Choice

Use the explicit formula to find the first four terms of the sequence. $$a_n=3+5\left(n-1\right)$$  

• 3, 15, 75, 375

• 3, 8, 13, 18

• 5, 8, 11, 14

• 5, 15, 45, 135

Multiple Choice

The difference between two terms in an arithmetic sequence is called ___________________.

• Formula

• Common Ratio

• Common Difference

• Common Factor

Multiple Choice

In the sequence 9, 18, 27, 36 and so on, what is the common difference?

• 5

• 7

• 9

• 11

Multiple Choice

What 3 terms come next in the sequence? 
144, 72, 36

• 36, 18, 9
• 18, 16, 9
• 18, 9, 4.5
• 4.5, 9, 16

Fill in the Blank

Determine the next term in the sequence 20, 8, -4, ...

-

Multiple Choice

Write the explicit formula for the sequence.

10, -2, -14, -26, ...

• $$a_n=10\cdot\left(-\frac{1}{5}\right)^{\left(n-1\right)}$$

• $$a_n=-2\cdot7^{\left(n-1\right)}$$

• $$a_n=10-8\left(n-1\right)$$

• $$a_n=10-12\left(n-1\right)$$

Multiple Choice

Find the explicit formula:
-23, -16, -9, -2, ...

• a_n = -23 + 7(n -1)
• a_n = -23 - 7(n -1)
• a_n = 7 - 23(n -1)
• a_n = 7 + 23(n -1)

Multiple Choice

Is the sequence arithmetic: -3, 3, -3, 3, ...

• Yes
• No

Multiple Choice

Find the 25th term of the sequence if a₁ = 5 and d = 0.5

• 18
• 17
• 15
• 14

Multiple Choice

What is the common difference?

• −16
• 84
• 16
• 0

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2 dollar summary.  

Write a two-dollar (or more) summary of the arithmetic and geometric sequences.  Each word is worth 10 cents.