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11/6/17

4.7 Sequences

(Recursive & Explicit Formulas)

Arithmetic sequences • are linear functions that have a domain of positive consecutive in which the difference between any two consecutive terms is equal. • is a list of terms separated by a common difference, (the added to each consecutive term in an arithmetic sequence.) • can be represented by formulas, either explicit or recursive, and those formulas can be used to find a certain term of the sequence.

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Recursive Formula

The following slides will answer these questions:

• What does recursive mean? • What is the math formula? • How do you write the formula given a situation? • How do you use the formula?

Definition of Recursive

• relating to a procedure that can repeat itself indefinitely

• repeated application of a rule

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Recursive Formula

A formula where each term is based on the term before it.

Formula: A(n) = A(n-1) + d, A(1) = ?

For each list of below, determine the next three numbers in the list. 1, 2, 3, 4, 5, ______, ______, ______7, 9, 11, 13, 15, ______, ______, ______10, 7, 4, 1, -2, ______, ______, ______2, 4, 7, 11, 16, ______, ______, ______1, -1, 2, -2, 3, -3, ______, ______, ______

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For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, ______, ______, ______10, 7, 4, 1, -2, ______, ______, ______2, 4, 7, 11, 16, ______, ______, ______1, -1, 2, -2, 3, -3, ______, ______, ______

For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, ______, ______, ______2, 4, 7, 11, 16, ______, ______, ______1, -1, 2, -2, 3, -3, ______, ______, ______

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For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, -5, -8, -11 2, 4, 7, 11, 16, ______, ______, ______1, -1, 2, -2, 3, -3, ______, ______, ______

For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, -5, -8, -11 2, 4, 7, 11, 16, 22, 29, 37 1, -1, 2, -2, 3, -3, ______, ______, ______

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For each list of numbers below, determine the next three numbers in the list.

1, 2, 3, 4, 5, 6, 7, 8 These are all examples of you 7, 9, 11, 13, 15, 17, 19, 21 using the 10, 7, 4, 1, -2, -5, -8, -11 recursive formula. As long as you 2, 4, 7, 11, 16, 22, 29, 37 know the previous 1, -1, 2, -2, 3, -3, 4, -4, 5 term and the common difference!

Recursive Formula:

A(n) = A(n-1) +d, A(1) = ?

A(n) is the nth term of the sequence

A(n-1) is the previous term to nth term d is the common difference

A(1) is the first term of the sequence

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Recursive Formula:

A(n) = A(n-1) +d, A(1) = ?

A(n) is the nth term of the sequence To write a recursive A(n-1) is theformula previous you term only toneed nth term to identify the first term, d is the commonA(1) and difference the common difference. A(1) is the first term of the sequence

Example: Write the recursive formula

Briana borrowed $870 from her parents for airfare to Europe. She will pay them back at the rate of $60.00 per month. Let A(n) be the amount she still owes after n months. • Describe in words: Start at $870 and add -60 each time. • The formula is : A(n) = A(n-1) + d, A(1) = ? A(n) = A(n-1) + -60, A(1) = 870

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Example: Write the recursive formula

If you buy a new car, you might be advised to have an oil change after driving 1,000 miles and every 3,000 miles thereafter. The following sequence gives the mileage when oil changes are required: 1000 4000 7000 10000 13000 16000

• Describe in words: Start at 1,000 and add 3,000 each time. • The formula is : A(n) = A(n-1) + d, A(1) = ? 10/6/14A(n) = A(n-1) + 3,000, A(1) = 1,000

Example: Write the recursive formula

Consider the sequence generated by 2000, 2040, 2080, 2120, 2160,. . .

• Describe this sequence in words: Start at 2000 and add 40 each time. • The formula is : A(n) = A(n-1) + d, A(1) = ?

A(n) = A(n-1) +40, A(1) = 2000

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Using the Recursive Formula:

• Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000

In order to find the 4th term, you need to find the 2nd term, 3rd term and then find the 4th term. (Basically using the formula three times!—using it recursively!)

Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 First Term = 2000 Second Term = 2000 + 40 = 2040 Third Term = 2040 + 40 = 2080 Fourth Term = 2080 + 40 = 2120

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Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(1) = 2000 A(2) = 2000 + 40 = 2040 A(3) = 2040 + 40 = 2080 A(4) = 2080 + 40 = 2120

Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = ? A(1) +40 2000 +40 2040

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Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(1) +40 2000 +40 2040

Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = ? A(2) +40 2040 +40 2080

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Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(2) +40 2040 +40 2080

Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(4) = A(4-1) +40, A(4) = ? A(3) +40 2080 +40 2120

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Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(4) = A(4-1) +40, A(4) = 2120 A(3) +40 2080 +40 2120

Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 First Term = 2000 Second Term = 2000 + 40 = 2040 Third Term = 2040 + 40 = 2080 Fourth Term = 2080 + 40 = 2120

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Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040

A(3) = A(3-1) +40, A(3) = 2080

A(4) = A(4-1) +40, A(4) = 2120

Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 Recursive: repeated A(3) = applicationA(3-1) +40, of a A(3) = 2080 rule. A(4) = A(4-1) +40, A(4) = 2120

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Recursive Formula:

Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 Is there an easier way to A(3) = A(3find-1) the +40, 4th A(3) = 2080 term? A(4) = A(4-1) +40, A(4) = 2120

Now find the 100th Term in each sequence

1, 2, 3, 4, 5, 6, 7, 8… 7, 9, 11, 13, 15, 17, 19, 21… Is the recursive 10, 7, 4, 1, -2, -5, -8, -11… formula a useful way to 2, 4, 7, 11, 16, 22, 29, 37… find the value 1, -1, 2, -2, 3, -3, 4, -4, 5… of the 100th term?

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Recursive Formula

Is the recursive formula efficient to finding the 2nd or 3rd term in a sequence? YES

Is the recursive formula efficient in finding the 75th term in a sequence? NO

Explicit Formula

The following slides will answer these questions: • What does explicit mean? • What is the math formula? • How do you write the formula given a situation? • How do you use the formula?

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Definition of Explicit

• stated clearly and in detail

• leaving no room for confusion or doubt

Explicit Formula:

• Formula where any term can be found by substituting the number of that term.

• Function rule that allows you to find any term, as long as you know which term you want to find

A(n) = A(1) + (n – 1)d

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Explicit Formula:

A(n) = A(1) + (n – 1)d

A(n) is the nth term of the sequence

A(1) is the first term of the sequence d is the common difference

Example: Write the explicit formula

If you buy a new car, you might be advised to have an oil change after driving 1,000 miles and every 3,000 miles thereafter. At what milage would you have your 5th oil change? • Describe this sequence in words: Start at 1000 and add 3,000 each time. A(n) = A(1) + (n – 1)d

A(5) = 1000 + (5 – 1)3000

th 10/6/14Your 5 oil change would be at 13,000 miles.

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Got it? #3:

On pg. 276, Complete Got it #3 a-d

Use the Explicit formula, NOT the recursive formula.

10/6/14

Got it? # 3 ANSWERS a) A(9) = 51 b) A(9) = 119 c) A(9) = 11.3 d) A(9) = 25

10/6/14

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Got it? #4:

On pg. 277, Complete Got it #4 a & b

10/6/14

Got it? # 4A ANSWER

A(n) = A(1) + (n – 1)d = 100 + (16– 1) (-1.75) = 100 + (15) (-1.75) = $73.75

* The Key word is AFTER 15 rides.

10/6/14

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Got it? # 4B ANSWER

100 1.75 = 57.14 or 57 rides

10/6/14

Arithmetic Sequences

Since you must know the first term and the common difference for both the explicit and recursive formula, you can write the formulas interchangeably.

Try it in Got it #5 & #6 on pg. 278

11/6/17

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Explicit vs. Recursive

• An explicit formula is a formula used to find the nth term of a sequence

• A recursive sequence is the process in which each step of a pattern is dependent on the step or steps before it. • A recursive formula is a formula used to find the next term of a sequence when the previous term is known.

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