Review on Sequence and Series

Multiple Choice

what is the 101th term in this sequence of 3 5 7 9 ?

Hint: A_n=A₁+(n-1)d

• 203

• 101

• 103

• 200

Sequences

A sequence is an ordered list of numbers. Each number in a sequence is called a term. Each term an has a specific position n in the sequence.

example: 5, 10, 15, 20, 25,…a_n

Arithmetic Sequence

An arithmetic sequence is a sequence where you add a constant number to find the next term, similar to the constant rate of change in a linear function.

In an arithmetic sequence we only add to find the next number (term). This number can be positive or negative.

Term

In a sequence each number is called a term.

The dots at the end mean the sequence continues forever.

a₁ is the first term

a₂ is the second term

a₃ is the third term and so on...

a_n is the nth term (we will find the equation for the nth term in the next lesson)

Common Difference

The number you add/subtract to find the next term in a sequence is called the common difference.

Subtract two terms to find the common difference. d is used to define the common difference

Ready to try some examples?

Remember in an arithmetic sequence, each number is found by adding a common difference, which can be positive or negative.

Multiple Choice

What is the lesson about?

• Arithmetic Sequences

• Geometric Sequences

• No idea

• Some math stuff

Multiple Choice

What is each number in a sequence called?

• common difference

• term

• sequence

• arithmetic

Multiple Choice

What is the difference between two numbers in an arithmetic sequence called?

• common difference

• term

• sequence

• number

Multiple Choice

Identify the common difference.

10, 20, 30, 40, ...

• 5

• 10

• 15

• 20

Multiple Choice

The next two terms of the sequence 12,17,22,27 are

• 28 and 33

• 29 and 34

• 32 and 37

• 30 and 35

Multiple Choice

Find the common difference: 29, 21, 13, 5, ...

Remember, the common difference is the difference between the numbers.

• d = -10

• d = -9

• d = -8

• d = -7

Multiple Select

Which sequences are arithmetic?

(hint: select 2)

• 2, 4, 6, 8, 10 ...

• 5, 10, 20, 40, 80, 100 ...

• 1, 3, 7, 9, 15, 20 ...

• 4, 8, 12, 16, 20, 24 ...

Multiple Choice

 The formula for finding the $$n^{th}$$  term of the arithmetic sequence:

15, 13, 11, 9 ...

* what is the first term?
* what is the common difference?

• $$15+\left(n-1\right)\left(2\right)$$  

• $$9+\left(n-1\right)\left(2\right)$$  

• $$15+\left(n-1\right)\left(-2\right)$$  

Multiple Choice

The first term of an arithmetic sequence is 2 while the 18th term is 87. Find the common difference of the sequence.

• 7

• 6

• 5

• 3

Multiple Choice

Which term of the arithmetic sequence 5,9,13,17,.....is 401?

• 99th term

• 100th term

• 111th term

• 112 term

ARITHMETIC SERIES

It is the sum of the terms of the given arithmetic sequence.

ARITHMETIC SERIES

ARITHMETIC SERIES

Fill in the Blanks

Find the sum : 8+12+16+20+24+28+32+36+40

Fill in the Blanks

Find the sum: 7+11+15+19+23+27+31+35

ARITHMETIC SERIES

ARITHMETIC SERIES

Fill in the Blanks

Find the specified partial sum: 3+9+15+...

$S_{24}$  

Fill in the Blanks

Find the specified partial sum of

$a_1=32;\ d\ =\ 4;\ S_{26}$  

​Arithmetic Series

Formula

Bebs Ever Tots​

Multiple Choice

What is the sum of the 1st 5 terms of $$a_n=3n+2$$  ?

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

• 55

• 22

• 5

• 2

Multiple Choice

Which of the following is the correct formula for finding the Arithmetic Series?

• $$S_n=2n\left(a_1\cdot a_n\right)$$

• $$S_n=2n\left(a_1+a_n\right)$$

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

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

Multiple Choice

Find S₁₄ 
7, 2, -3, ...

• -812
• 119
• -945
• -357

​Geometric Sequences are very important in investing/saving your money

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Let's learn more​

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Two different ways to show the same thing

Some text here about the topic of discussion

​Explicit Formua

​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

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

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​Recursive Formula

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

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r = common ratio

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n - 1 = previous term​

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n = desired term​

​

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​Some text here about the topic of discussion.

​Explicit Formula

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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 the explicit formula?

• a_n = 6*(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

Some text here about the topic of discussion

Common ratio less than 1​

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64, 32, 16, 8, 4, ...​

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r = 1/2​

Decay

Common ratio greater than 1

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4, 8, 16, 32, 64, ...​

​

r = 2​

Growth

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

Some text here about the topic of discussion

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

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

GEOMETRIC SERIES

The sum of the terms in a geometric sequence.

GEOMETRIC SERIES

GEOMETRIC SERIES

Multiple Choice

Is the sequence 5,15,45,... a geometric sequence?

• Yes, it is a geometric sequence with r=3.

• No, it is not a geometric sequence.

Multiple Choice

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

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

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

Multiple Choice

Evaluate the geometric series $$1-2+4-8,\ ...,\ \ \ \ \ \ \ n=8$$  

• $$\frac{1}{3}$$  

• -86

• -87

• -85

​Example 1: Find the sum of the series

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4 + 12 + 36 + 108 + 324 + ​972 + 2916

​Example 2: Find the sum of the series

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-15 + 30 - 60 + 120 - 240 + 480​

​a₁ = - 15

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r = -2

​

n = 6​

​Example 3: Find the sum

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​Example 3

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Does the series converge or diverge? If it converges, find its sum

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Multiple Choice

Does the following series converge or diverge?

$$\frac{1}{4}-\frac{3}{8}+\frac{9}{16}-\frac{27}{32}+.\ .\ .$$  

• converge

• diverge

Multiple Choice

Find the sum of the infinite geometric series, if it exists. $$\sum_{n=1}^{\infty}8\left(\frac{1}{5}\right)^{n-1}$$  

• 8

• 8/5

• 10

• Does not exist

Multiple Choice

$$\sum_{k=1}^{\infty}\left(\frac{4}{5}\right)^{k-1}$$  Find the sum of the series if it exists.

• 4/5

• 5

• 6.5

• Does not exist

Multiple Choice

Determine whether the sequence is convergent or divergent:

a_n=2 (⅓)^n-1

• Convergent

Multiple Choice

What best describes the following infinite geometric series:

1/2 + 3/4 + 9/8 + ...

• Diverges

• Converges