# Common Core: High School - Functions : Sequences as Functions: CCSS.Math.Content.HSF-IF.A.3

## Example Questions

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### Example Question #1 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

Based on the following sequence, what is the value of the triangle?

Explanation:

This question is testing ones ability to recognize sequences as functions.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

For this particular problem, the common difference needs to be calculated as it is an arithmetic sequence.

To identify the pattern or in other words, calculate the common difference, subtract the first term from the second term. Then subtract the second term from the third term. For an arithmetic sequence these two differences should be equal to one another.

Given this particular sequence

the common difference using the above method is as follows.

Step 2: Find the value of the square.

To find the value of the square add the common difference to the previous term in the sequence.

The term before the square is nine, therefore the value of the square is,

.

The sequence is now,

Step 3: Find the value of the triangle.

To find the value of the triangle add the common difference to the previous term in the sequence.

### Example Question #2 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

What are the next two values in the following sequence?

Explanation:

This question is testing ones ability to recognize sequences as functions.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

For this particular problem, the common ratio needs to be calculated as it is an geometric sequence.

To identify the pattern or in other words, calculate the common ratio, divide the second term by the first term. Then divide the third term by the second term. For an geometric sequence these two ratios should be equal to one another.

Given this particular sequence

the common ratio using the above method is as follows.

Step 2: Find the next value.

To find the value of the next term multiply the common ratio to the previous term in the sequence.

The last term is 49, therefore the value of the next term is,

.

The sequence is now,

Step 3: Find the next value.

To find the value of the next term multiply the common ratio to the previous term in the sequence.

The next two terms in the sequence are, .

### Example Question #3 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

Find the 26th term in the following sequence.

Assume the sequence starts with input of one.

Explanation:

This question is testing ones ability to recognize sequences as functions and identify specific entry values.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

Since the sequence starts with the assumed input value of zero and the given sequence values of,

the following logic statement can be created.

Let  represent the sequence value for the input value . In other words,

Looking at the values, the difference between entry two and entry one is two. The difference between entry three and entry two is four. The difference between entry four and entry three is eight. Finally, the difference between entry five and entry four is sixteen. This pattern signifies that the increase between each entry is double the difference of the previous terms.

Step 2: Write the logic statement for the sequence in mathematical terms.

Step 3: Find the function for this particular sequence.

Since the previous term minus the term before it will always equal two, another way to write this sequence is

Step 3: Verify function for known terms.

Step 4: Calculate the specific value in question.

### Example Question #4 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

What is the value of the square in the following sequence?

Explanation:

This question is testing ones ability to recognize sequences as functions and identify specific entry values.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

Looking at the above sequence it is seen that each term is a perfect square. Perfect square are composed of one number being multiplied by itself. In other words, when one takes the square root of a perfect square the value that is found is a factor that when squared results in the perfect square.

In mathematical terms this concept looks like the following.

Step 2: Write the sequence with the identified pattern found in step 1.

Becomes,

.

Step 3: Continuing the pattern, solve for the triangle and square.

Therefore,

### Example Question #5 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

What are the missing values in the following sequence.

Explanation:

This question is testing ones ability to recognize sequences as functions.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

For this particular problem, the common difference needs to be calculated as it is an arithmetic sequence.

To identify the pattern or in other words, calculate the common difference, subtract the first term from the second term. Then subtract the second term from the third term. For an arithmetic sequence these two differences should be equal to one another.

Given this particular sequence the common difference is calculated as follows.

Therefore, the common difference is negative six. In other words, each term decreases by six every time.

Step 2: Use the common difference to find the missing terms in the sequence.

The sequence becomes,

therefore.

.

### Example Question #6 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

What is the next term in the following sequence?

Explanation:

This question is testing ones ability to recognize sequences as functions.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

For this particular problem, the common difference needs to be calculated as it is an arithmetic sequence.

To identify the pattern or in other words, calculate the common difference, subtract the first term from the second term. Then subtract the second term from the third term. For an arithmetic sequence these two differences should be equal to one another.

Given this particular sequence

Therefore, the common difference for this sequence is negative four. In other words, the sequence is decreased by four every time.

Step 2: Calculate the next term in the sequence by adding the common difference to the last term given.

### Example Question #7 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

What is the ninth term in the following sequence.

Assume the first entry comes from the input value of zero.

Explanation:

This question is testing ones ability to recognize sequences as functions and identify specific entry values.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

Since the sequence starts with the assumed input value of zero and the given sequence values of,

the following logic statement can be created.

Let  represent the sequence value for the input value . In other words,

Looking at the values, the difference between entry one and entry zero is one. The difference between entry two and entry one is two. The difference between entry three and entry two is four. The difference between entry four and entry three is eight. Finally, the difference between entry five and entry four is sixteen. This pattern signifies that the increase between each entry is double the entry of the previous term.

Step 2: Write the logic statement for the sequence in mathematical terms.

Since  is equaled to a value other than zero, it is known that  must exist as an exponent.

where  and  are some constants.

Step 3: Apply the function to the above sequence to verify.

This shows that

thus,

Step 4: Use the function for the sequence to calculate the specific entry value.

### Example Question #8 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

Given that the Fibonacci sequence is

for all

.

What is the sequence for the first eight entries.

Explanation:

This question is testing ones ability to recognize sequences as functions. It is also testing the concept of what it means for a function to be recursive. Recall that a function is recursive when it requires the repeat process to find the next term in a sequence.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

The approach in this particular case.

I. Use the function notation for the Fibonacci sequence. Recalling that the Fibonacci sequence is defined as adding the current term with the previous term to result in the next term.

In mathematical terms this is,

for all

II. Continuing the pattern found, adding each term until the twelfth term is found.

Step 2: Continue the pattern to find the particular term.

For this particular case we want to find the first eight terms of the sequence,

The sequence for the first eight entries

### Example Question #9 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

Given the following sequence, find the next term.

Explanation:

This question is testing ones ability to recognize sequences as functions.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

For this particular problem, the common difference needs to be calculated as it is an arithmetic sequence.

To identify the pattern or in other words, calculate the common difference, subtract the first term from the second term. Then subtract the second term from the third term. For an arithmetic sequence these two differences should be equal to one another.

Given this particular sequence the common difference is calculated as follows.

Therefore, the common difference is negative three. In other words, each term decreases by three every time.

Step 2: Use the common difference to find the missing terms in the sequence.

The next term in the sequence is .

### Example Question #10 : Sequences As Functions: Ccss.Math.Content.Hsf If.A.3

What is the correct function that describes the following sequence?

Assume the sequence starts with the input value of zero.

Explanation:

This question is testing ones ability to recognize sequences as functions.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the pattern of the given sequence.

Since the sequence starts with the assumed input value of zero and the given sequence values of,

the following logic statement can be created.

Let  represent the sequence value for the input value . In other words,

Since  is equaled to a value other than zero, it is known that  must exist as an exponent.

Step 2: Write the general formula of the sequence.

where  and  are some constants.

Step 3: Using the formula from step 2 and the known characteristics from step 1, find the function that describes the sequence.

Since any value raised to the zero power equals one, the following function can be simplified and the constant  can be solved for.

Substituting the value for  into the function, solve for .

Since any value raised to the power of one equals the number itself, the following function can be simplified to solve for .

Substitute the value for  into the function to get the final solution.

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