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When studying math, it''s useful to know how to express patterns in function notation. Patterns appear all throughout nature. In many fields, it''s important to know how to express them in a formal mathematical fashion. If we''re studying a plant that grows in a specific pattern, we might want to know how to mathematically express that pattern. Before we learn how to write number patterns in function notation, it''s a good idea for us to look over closely related topics such as functions and input-output tables. These concepts will help us understand how to write number patterns in function notation.

If we have a pattern of numbers, we might want to represent it in the form of a function. Once we have the pattern in the form of a function, we can generate the pattern to as many terms as we wish. An arithmetic sequence is a sequence that changes by a constant value with each consecutive term. Consider the following sequence:

2, 10, 18, 26, 34, 42, 50, 58…

This sequence is arithmetic because it increases by a constant value of eight. This is all well and good, but what if we needed to know the 386

f(n) = a_{1} + (n-1)d

Let''s try this formula out on the sequence above.

We know that a

f(n) = 2 + (n-1)8

Distribute the eight.

f(n) = 2 + 8n – 8

f(n) = 8n – 6

f(n) = 8n – 6

Now we have an all-purpose function that we can use to generate any term of this sequence. If we want the fourth term of the sequence, we can say that:

f(4) = 8(4) -6 = 26

Looking at the sequence above, we see that this is correct. If we want the 386

f(386) = 8(386) – 6 = 3082

We also have geometric sequences. These sequences differ by a common ratio with each term. Consider the sequence below.

2, 8, 32, 128, 512, 2048, 8192, 32768…

This is a geometric sequence because we multiply each term by a common ratio of four to get the next term. In order to create a function to find the n

f(n) = a_{1} * r^{n-1}.

In this function, a

f(n) = 2 * 4^{n-1}

Let''s use this formula to calculate the seventh value of the sequence.

f(7) = 2 * 4^{7-1}

f(7) = 2 * 4^{6}

f(7) = 2 * 4096

f(7) = 8192.

f(7) = 2 * 4

f(7) = 2 * 4096

f(7) = 8192.

As we can see from the above sequence, this answer checks out.

Now that we know the concepts, let''s do some practice problems. We should do each of these calculations on our own before looking at the answers.1. Write the following pattern in function notation.

6, 9, 12, 15, 18, 21, 24, 27…

Since this increases by a constant value of three, it''s an arithmetic sequence and we can use the following formula:

f(n) = a_{1} + (n-1)d

We know that

a_{1} = 6

and d = 3

, so we plug these in.f(n) = 6 + (n-1)3

Distribute the three.

f(n) = 6 + 3n -3

f(n) = 3n +3

f(n) = 3n +3

2. Write the following pattern in function notation.

5, 3, 1, -1, -3, -5, -7, -9…

Since this sequence decreases by two with each term, we know that it''s arithmetic.

f(n) = a_{1} + (n-1)d

a_{1} = 5

d = -2

f(n) = 5 +(n-1)(-2)

a

d = -2

f(n) = 5 +(n-1)(-2)

Distribute the -2.

f(n) = 5 + -2n +2

f(n) = -2n + 7

f(n) = -2n + 7

3. Write the following pattern in function notation.

5, 10, 20, 40, 80, 160, 320, 640…

This sequence changes by a ratio of two. We multiply each term by two to get the following term. Therefore, we use the formula for a geometric sequence.

f(n) = a_{1} * r^{n-1}.

a_{1 }= 5

r = 2

a

r = 2

Plug these in.

f(n) = 5 * 2^{n-1}.

4. Write the following sequence in function notation and use the function to calculate the 11th term.

9, 27, 81, 243, 729, 2167, 6561, 19683…

Each term in this sequence increases by a ratio of three. This is a geometric sequence and we need to use the corresponding formula.

f(n) = a_{1} * r^{n-1}.

a_{1} = 9

r = 3

a

r = 3

Plug these values in.

f(n) = 9 * 3^{n-1}.

Now we need to use this to calculate the 11th term.

f(11) = 9 * 3^{11-1 }^{}f(11) = 9 * 3^{10}^{}f(11) = 531,441

5. Write the following sequence in function notation and use the function to calculate the 67

13, 16, 19, 22, 25, 28, 31, 34…

This pattern increases by a constant value of three, so we know that it''s arithmetic.

f(n) = a_{1} + (n-1)d

a_{1} = 13

d = 3

f(n) = 13 + (n-1)3

f(n) = 13 +3n -3

f(n) = 3n + 10

a

d = 3

f(n) = 13 + (n-1)3

f(n) = 13 +3n -3

f(n) = 3n + 10

Now we calculate the 67th value.

f(67) = 3(67) + 10

f(67) = 211

f(67) = 211

6. Write the following sequence in function notation and use the function to calculate the 13th term.

4, 8, 16, 32, 64, 128, 256, 512…

Since this pattern increases by a factor of two, we know that it''s geometric and we must use the corresponding formula.

f(n) = a_{1} * r^{n-1}^{}a_{1} = 4

r = 2

f(n) = 4 * 2^{n-1}

r = 2

f(n) = 4 * 2

Now we''ll calculate the thirteenth term of this sequence.

f(13) = 4 * 2^{13-1}^{}f(13) = 4 * 2^{12}^{}f(13) = 16384

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