Sequences

Mathematical Patterns

Suppose each student in your math class has a phone conversation with every other member of the class. What is the minimum number of calls required?

Instead of actually making the calls, you can represent telephone conversations by drawing like the ones below.

?

How many calls are necessary for two people to have a conversation?

How many calls are necessary for everyone to talk to everyone else in a group of three people? In a group of four people?

Use a diagram to find the number of calls needed for five people?

Which of the following expressions represent the pattern for number of telephone calls?

How many calls would be needed for this class?

€

2n − 3

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n(n −1) − 5

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n(n −1)

2

Vocabulary

• Sequence – an ordered list of numbers that can be described by a pattern.

• Term – any number in the sequence

1st term 2nd term 3rd term … n - 1 term nth term n + 1 term

a1 a2 a3 … an-1 an an+1

Variables can be used to represent terms of a sequence: a is typically used.

Recursive Formula

• Defines the terms of the sequence by relating each term to the term before it.

{2, 4, 6, 8, 10…} {0, 1, 3, 6, 10, 15}

Write a recursive formula for each sequence.*Explicit formulas do not require the use or knowledge of prior terms

Stacking Boxes

You are stacking boxes in levels that form squares. The numbers of boxes in successive levels form a sequence. The figure shows the top five levels.• How many boxes of equal size would you

need for the next lower level?• How many boxes of equal size would you

need to add three levels?

Arithmetic Sequences

• The difference between consecutive terms in the sequence is constant.

• This constant is called the COMMON DIFFERENCE

6, 12, 18, 24 … What is the common

difference?

Arithmetic Sequence Formulas

Recursive Formula

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an = an−1 + d

Explicit Formula

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an = a1 + (n −1)d

n is the term being described

d is the common difference

Arithmetic Mean

• Average between two values

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value 1 + value 2

2

Question: a2 of an arithmetic sequence has a value of 6. a4 of the same sequence has a value of 12. What is the value of a3?

Geometric Sequence

• The ratio between consecutive terms is constant.

Question: Why is it called a common ratio (while in arithmetic sequences, it is called a common difference)?

• Find the 6th term (or in this case, shape) in the sequence.

• Write a formula that predicts how many triangles will be present in the nth term.

Geometric Sequence Formulas

Recursive Formula

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an = an−1( )r

Explicit Formula

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an = a1rn−1

n is the term being described

r is the common ratio

Geometric Mean

• Geometric Mean between two positive values:

Question: The second term of a geometric sequence has a value of 8. The fourth term of the same sequence has a value of 18. What is the value of the third term?

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value1( ) value 2( )

The golferA particular golfer, sadly misses each putt on a

particularly bad hole. The ball ends up halfway beyond the distance it started away from the cup.

Write a sequence to represent the ball’s distance from the hole after each putt if the ball starts 12 feet from the hole.

Draw a diagram showing the result after each putt.