Writing Recursive and Closed Formulas for Simple Arithmetic and Geometric Sequences

WRITING RECURSIVE AND CLOSED FORMULAS FOR

SIMPLE ARITHMETIC AND GEOMETRIC SEQUENCES

An Interactive Tutorial by S. Mahaffey (Osborne High School)

FACTS ABOUT SEQUENCES…

Sequences are functions. They are made up of terms that go in a specific order.

The position of each term in the sequence is called n. For the 5th term, n = 5. For the 8th term, n = 8.

Each term is called an (pronounced “a sub n”).

SEQUENCE NOTATION…Consider the sequence 2, 4, 6, 8, …

The first term is called a1, “a sub 1”. For this example, a1 = 2.

The second term is 4, so a2 = 4.

The third term is 6, so a3 = 6.

The fourth term is 8, so a4 = 8.

The nth term is called an.

LET’S TRY A FEW EXAMPLES…

Consider the sequence 15, 20, 25, 30, 35, …

What is a1? 15 20 25 30 35

What is a3? 15 20 25 30 35

What is a5? 15 20 25 30 35

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DOMAIN AND RANGE The domain of a sequence is it’s n values.

Since n is just the position of the term, then the domain is the whole numbers 1, 2, 3, 4, etc.

The range of a sequence is it’s an values, or terms.

LET’S PRACTICE…Which is the correct domain and range for the

sequence 3, 10, 17, 24, …

D:{1,2, 3, 4,…} R:{3, 10, 17, 24,…}

D:{3, 10, 17, 24,…} R:{1, 2, 3, 4, …}

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SAME, BUT DIFFERENT… Sometimes the terms of the sequence

are not listed. Instead, you are given an equation.

If this is the case, you can find the first term by letting n=1, and the second term by letting n=2, and so on.

EXAMPLE: Find the domain and range of the sequence an =

3(n-1) + 2.

First, let’s write out the first 4 terms:a1 = 3(1-1) + 2 = 3(0) + 2 = 2a2 = 3(2-1) + 2 = 3(1) + 2 = 5a3 = 3(3-1) + 2 = 3(2) + 2 = 8a4 = 3(4-1) + 2 = 3(3) + 2 = 11

So, the domain is {1, 2, 3, 4, …} and the range is {2, 5, 8, 11, …}

YOUR TURN Find the domain and range for an = 2(n-1) + 10

D: {1, 2, 3, 4, …} R: {2, 4, 6, 8, …}

D: {1, 2, 3, 4, …} R: {10, 20, 30, 40, …}

D: {1, 2, 3, 4, …}R: {10, 12, 14, 16, …}

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YOUR TURN Find the domain and range for an = -4(n-1) + 1

D: {1, 2, 3, 4, …} R: {-2, -4, -6, -8, …}

D: {1, 2, 3, 4, …} R: {1, -3, -7, -11, …}

D: {1, 2, 3, 4, …}R: {-4, -8, -16, -32, …}

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YOUR TURN Find the domain and range for

an = 5∙2(n-1)

D: {1, 2, 3, 4, …} R: {5, 10, 20, 40, …}

D: {1, 2, 3, 4, …} R: {5, 10, 15, 20, …}

D: {1, 2, 3, 4, …}R: {1, 10, 100, 1000, …}

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WRITING FORMULAS… In the last 3 examples, the sequences

were given as formulas and you had to calculate the terms.

Now, we’re going to focus on how to write the formulas when you are given the terms.

RECURSIVE AND CLOSED There are two types of equations we’ll

be looking at in this tutorial.

The first type of formula is called a recursive formula.

The second type of formula is called a closed formula.

RECURSIVE FORMULAS… Recursive formulas are the easiest to write. Each term is defined by the term that comes before it. (The previous term).

Recursive formulas have 2 parts: define the first term define the next term by its relationship to the previous term.

For example, in the sequence 90, 95, 100, 105, … the first term is 90 and the next term is five plus the previous term.

In the sequence 14, 12, 10, 8, … the first term is 14 and the next term is two minus the previous term.

In the sequence 1, 2, 4, 8, 16, … the first term is 1 and the next term is 2 times the previous term.

WRITING RECURSIVE FORMULAS…

EnglishMath Symbol

The next term an

The previous term an-1

Recursive formulas are written almost exactly like you’d say it in English.

But instead of writing out phrases like “the next term”, you’ll use these math symbols instead:

LET’S PRACTICE… Write the recursive equation for the series

9, 11, 13, 15, …

Identify the pattern: The first term is 9The next term is equal to the previous term plus 2.

Write the equation:a1 = 9an = an-1 + 2

LET’S PRACTICE… Write the recursive equation for the series 25,

30, 35, 40, 45…

Identify the pattern: The first term is 25. The next term is equal to the previous term plus 5.

Write the equation:a1 = 25an = an-1 + 5

LET’S PRACTICE… Write the recursive equation for the series 10,

9, 8, 7, 6…

Identify the pattern: The first term is 10.The next term is equal to the previous term minus 1.

Write the equation:a1 = 10an = an-1 - 1

LET’S PRACTICE… Write the recursive equation for the series 3, 6,

12, 24, 48…

Identify the pattern: The first term is 3. The next term is equal to the previous term times 2.

Write the equation:a1 = 3an = an-1 ∙ 2

You try: Choose the correct recursive formula for the

sequence 3, 7, 11, 15, 19, …

a1 = 3 a1 = 3an = an+1 + 4 an = an-1 + 4

a1 = 4 a1 = 4an = an+1 + 4 an = an-1 + 4

You try: Choose the correct recursive formula for the

sequence 19, 15, 11, 7, 3, …

a1 = 19 a1 = 4an = an-1 + 4 an = an-1 + 19

a1 = 19 a1 = -4an = an-1 - 4 an = an-1 - 19

You try: Choose the correct recursive formula for the

sequence 7, 14, 28, 56, 112, …

a1 = 7 a1 = 7an = an-1 ∙ 2 an = an-1 + 7

a1 = 2 a1 = 2an = an-1 + 7 an = an-1 ∙ 7

You try: Choose the correct recursive formula for the

sequence 1, 8, 64, 512, …

a1 = 8 a1 = 1an = an-1 ∙ 8 an = an-1 + 82

a1 = 8 a1 = 1an = an-1 ∙ 1 an = an-1 ∙ 8

WE’RE ALMOST DONE…Recursive formulas are easy to write, but if you want

to find the 50th term, a50, you have to know the first 49 terms. This can be time consuming.

With closed formulas, you can easily find the 50th term (n = 50). The formula uses the variable n instead of an-1 (which is the value of the previous term).

So, if you want the 10th term, then n=10. If you want the 20th term, then n=20…

WRITING CLOSED FORMULAS FOR ARITHMETIC SEQUNCES:

When you add or subtract to get to the next term, then you have an arithmetic sequence:

To write a closed formula, start with the recursive formula:

a1 = an = an-1 +

Then rearrange the equation to get the closed formulas:

an = (n-1) +

FOLLOW THE PATTERN:Arithmetic:

Recursivea1 = 3an = an-1 + 4

Closedan = 4 (n-1) + 3

LET’S TRY AN EXAMPLE:

Find the 101st term of the sequence 2, 6, 10, 14, …

First write the recursive formula (The first term is 2 and each additional term is the previous term plus 4)a1 = 2an = an-1 + 4

Next, write the closed formula: an = 4(n-1) + 2

Finally, find the 101st term:a101 = 4(101-1) + 2 = 4(100) + 2 = 402

LET’S TRY AN EXAMPLE:

Find the 51st term of the sequence 12, 15, 18, 21, …

First write the recursive formula (The first term is 12 and each additional term is the previous term plus 3)a1 = 12an = an-1 + 3

Next, write the closed formula: an = 3(n-1) + 12

Finally, find the 51st term:a101 = 3(51-1) + 12 = 3(50) + 12 = 162

LET’S TRY AN EXAMPLE:

Find the 81st term of the sequence 100, 90, 80, 70, …

First write the recursive formula (The first term is 100 and each additional term is the previous term minus 10)a1 = 100an = an-1 - 10

Next, write the closed formula: an = -10(n-1) + 100

Finally, find the 81st term:a101 = -10(81-1) + 100 = -10(80) + 100 = -700

YOUR TURN… Write the closed formula for the

sequence: a1 = 3 an = an-1 + 8

an = 8(an-1) + 3 an = 3(an-1) + 8

an = 8(n-1) + 3 an = 3(n-1) + 8

YOUR TURN… Write the closed formula for the

sequence: a1 = 15 an = an-1 - 2

an = 2(n-1) + 15 an = -2(n-1) + 15

an = -2(n-1) - 15 an = 2(n-1) - 15

YOUR TURN… Write the closed formula for the

sequence: 15, 20, 25, 30, …

an = 15(n-1) - 5 an = 15(n-1) + 5

an = 5(n-1) - 15 an = 5(n-1) + 15

YOUR TURN…Find the 101st term of the sequence:

8, 11, 14, 17, 20, …

a101 = 803 a101 = 308

a51 = 158 a100 = 305

YOUR TURN…Find the 51st term of the sequence:

81, 79, 77, 75, 73, …

a101 = -19 a101 = -119

a51 = -19 a51 = -119

WRITING CLOSED FORMULAS FOR GEOMETRIC SEQUNCES:

When you multiply or divide to get to the next term, then you have a geometric sequence:

To write a closed formula, start with the recursive formula:

a1 = an = an-1 ∙

Then rearrange the equation to get the closed formulas:

an = ∙ (n-1)

FOLLOW THE PATTERN:Geometric:

Recursivea1 = 3an = an-1 ∙ 4

Closedan = 3 ∙ 4 (n-1)

LET’S TRY AN EXAMPLE:

Find the 9th term of the sequence 1, 2, 4, 8, 16, …

First write the recursive formula (The first term is 1 and each additional term is the previous term times 2)a1 = 1an = an-1 ∙ 2

Next, write the closed formula: an = 1 ∙ 2 (n-1)

Finally, find the 9th term:a9 = 1 ∙28 = 1∙256 = 256

LET’S TRY AN EXAMPLE:

Find the 10th term of the sequence 2, 6, 18, 54, …

First write the recursive formula (The first term is 2 and each additional term is the previous term times 3)a1 = 2an = an-1 ∙ 3

Next, write the closed formula: an = 2 ∙ 3(n-1)

Finally, find the 10th term:a10 = 2 ∙ 39 = 2 ∙19,683 = 39,366

YOUR TURN… Write the closed formula for the

sequence: a1 = 3 an = an-1 ∙ 8

an = 3 ∙ 8(an-1) an = 8 ∙ 3(n-1)

an = 3 ∙ 8(n-1) an = 24(n-1)

YOUR TURN… Write the closed formula for the

sequence: a1 = 15 an = an-1 ∙ 2

an = 2 ∙15 an = 15 ∙ 2(n-1)

an = 30(n-1) an = 2(n-1) ∙ 15

YOUR TURN… Write the closed formula for the

sequence: 5, 15, 45, 135, …

an = 3 ∙5(n-1) an = 3 ∙5(n-1)

an = 5 ∙3(n-1) an = 5 ∙ 3(n-1)

YOUR TURN…Find the 8th term of the sequence:

1, 3, 9, 27, 81, …

a8 = 2,187 a7 = 729

a8 = 6,561 a7 = 2,187

YOUR TURN…Find the 10th term of the sequence:

2, 10, 50, 250, 1250, …

a10 = 1,953,125 a9 = 1,953,125

a10 = 3,906,250 a9 = 781,250