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

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Question

What is the twelfth term in the Fibonacci sequence,

Answer

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 $n\geq1$

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 have six terms of the sequence,

continuing the pattern results in the following.

$\f(5)=5 \f(6)=8 \f(7)=f(6)+f(5)=8+5=13 \f(8)=f(7)+f(6)=13+8=21 \f(9)=f(8)+f(7)=21+13=34 \f(10)=f(9)+f(8)=21+34=55 \f(11)=f(10)+f(9)=55+34=89 \f(12)=f(11)+f(10)=89+55=144$

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Question

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

$1, 5, 9, \square, \triangle$

Answer

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.

$1, 5, 9, \square, \triangle$

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.

$\a_1,a_2,a_3,... \a_2-a_1=d \a_3-a_2=d$

Given this particular sequence

$1, 5, 9, \square, \triangle$

the common difference using the above method is as follows.

$\a_1,a_2,a_3,... \5-1=4 \9-5=4 \ \textup{Common Difference}=4$

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.

$1, 5, 9, \square, \triangle$

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

$9+4=13\Rightarrow \square=13$ .

The sequence is now,

$1, 5, 9, 13, \triangle$

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.

$13+4=17\Rightarrow \triangle=17$

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Question

What are the next two values in the following sequence?

Answer

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.

$\a_1,a_2,a_3,... \ \ \frac{a_2}{a_1}=d \ \ \frac{a_3}{a_2}=d$

Given this particular sequence

the common ratio using the above method is as follows.

$\a_1,a_2,a_3,... \ \ \frac{7}{1}=7 \ \ \frac{49}{7}=7 \ \textup{Common Difference}=7$

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,

$49\times 7=343$ .

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.

$343\times 7=2401$

Step 4: Answer the question.

The next two terms in the sequence are, .

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Question

What is the correct function that describes the following sequence?

Assume the sequence starts with the input value of zero.

Answer

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,

$\f(0)=3 \f(1)=4 \f(2)=6 \f(3)=10 \f(4)=18 \f(5)=34$

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.

$\f(0)=a^0+b=3 \f(1)=a^1+b=4 \f(2)=a^2+b=6$

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

$\f(0)=a^0+b \f(0)=1+b \1+b=3 \b=2$

Substituting the value for into the function, solve for .

$\f(n)=a^n+2 \f(1)=a^1+2 \4=a^1+2$

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

$\4=a+2 \2=a$

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

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Question

Find the 26th term in the following sequence.

Assume the sequence starts with input of one.

Answer

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,

$\f(1)=0 \f(2)=2 \f(3)=6 \f(4)=14 \f(5)=30$

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.

$\f(3)=2(2-0)+2 \f(3)=4+2 \f(3)=6$

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.

$\f(1)=2^1-2=0 \f(2)=2^2-2=4-2=2 \f(3)=2^3-2=8-2=6 \f(4)=2^4-2=16-2=14 \f(5)=2^5-2=32-2=30$

Step 4: Calculate the specific value in question.

$\ f(26)=2^{26}-2 \f(26)=67108864-2 \f(26)=67108862$

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Question

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

$225,196,169,\triangle, 121, \square$

Answer

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.

$225,196,169,\triangle, 121, \square$

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.

$n=a^2\Rightarrow \sqrt{n}=a$

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

$225,196,169,\triangle, 121, \square$

Becomes,

$15^2,14^2,13^2,\bigtriangleup, 11^2, \square$ .

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

$15^2,14^2,13^2,\bigtriangleup, 11^2, \square\Rightarrow 15^2,14^2,13^2,12^2, 11^2,10^2$

Therefore,

$\ \triangle=12^2=144 \ \square=10^2=100$

Step 4: Answer the question.

$\square=100$

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Question

Which function correctly describes the following sequence.

Assume the sequence starts with the input value of zero.

Answer

his 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,

$\f(0)=4 \f(1)=6 \f(2)=12 \f(3)=30 \f(4)=84$

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.

$\f(0)=a^0+b=4 \f(1)=a^1+b=6 \f(2)=a^2+b=12$

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

$\f(0)=a^0+b \f(0)=1+b \1+b=4 \b=3$

Substituting the value for into the function, solve for .

$\f(n)=a^n+3 \f(1)=a^1+3 \6=a^1+3$

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

$\6=a+3 \3=a$

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

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Question

Given that the Fibonacci sequence is

for all

$n\geq1$ .

What is the sequence for the first eight entries.

Answer

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 $n\geq1$

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,

$\f(1)=1 \f(2)=f(1)+f(0)=1 \f(3)=f(2)+f(1)=1+1=2 \f(4)=f(3)+f(2)=2+1=3 \f(5)=f(4)+f(3)=3+2=5 \f(6)=f(5)+f(4)=5+3=8 \f(7)=f(6)+f(5)=8+5=13 \f(8)=f(7)+f(6)=13+8=21$

Step 3: Answer the question.

The sequence for the first eight entries

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Question

What are the missing values in the following sequence.

$150,144,138, \bigtriangleup, 126, \square$

Answer

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.

$150,144,138, \bigtriangleup, 126, \square$

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.

$\a_1,a_2,a_3,... \a_2-a_1=d \a_3-a_2=d$

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

$\150,144,138,... \144-150=-6 \138-144=-6$

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.

Step 3: Answer the question.

The sequence becomes,

therefore.

$\ \bigtriangleup=132 \\square=120$ .

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Question

Given the following sequence, find the next term.

Answer

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.

$\a_1,a_2,a_3,... \a_2-a_1=d \a_3-a_2=d$

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

$\12,9,6,3,0,-3 \9-12=-3 \6-9=-3$

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.

Step 3: Answer the question.

The next term in the sequence is .

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Question

What is the next term in the following sequence?

Answer

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.

$\a_1,a_2,a_3,... \a_2-a_1=d \a_3-a_2=d$

Given this particular sequence

$\16,12,8,... \12-16=-4 \8-12=-4$

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.

$\ \textup{Common Difference}=-4 \\textup{Last Term Known}=-4 \\textup{Next Term}=\textup{Last Term Known}+\textup{Common Difference} \\textup{Next Term}=-4+-4=-8$

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Question

What is the ninth term in the following sequence.

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

Answer

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,

$\f(0)=1 \f(1)=2 \f(2)=4 \f(3)=8 \f(4)=16\f(5)=32$

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.

$\f(0)=1 \f(1)=2\cdot f(0)=2\cdot 1=2 \f(2)=2\cdot f(1)=2\cdot 2=4 \f(3)=2\cdot f(2)=2\cdot 4=8 \f(4)=2\cdot f(3)=2\cdot 8=16\f(5)=2\cdot f(4)=2\cdot 16=32$

This shows that

thus,

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

$\f(9)=2\cdot f(8) \f(9)=2^9 \f(9)=512$

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Question

What is the twelfth term in the Fibonacci sequence,

Answer

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 $n\geq1$

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 have six terms of the sequence,

continuing the pattern results in the following.

$\f(5)=5 \f(6)=8 \f(7)=f(6)+f(5)=8+5=13 \f(8)=f(7)+f(6)=13+8=21 \f(9)=f(8)+f(7)=21+13=34 \f(10)=f(9)+f(8)=21+34=55 \f(11)=f(10)+f(9)=55+34=89 \f(12)=f(11)+f(10)=89+55=144$

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Question

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

$1, 5, 9, \square, \triangle$

Answer

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.

$1, 5, 9, \square, \triangle$

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.

$\a_1,a_2,a_3,... \a_2-a_1=d \a_3-a_2=d$

Given this particular sequence

$1, 5, 9, \square, \triangle$

the common difference using the above method is as follows.

$\a_1,a_2,a_3,... \5-1=4 \9-5=4 \ \textup{Common Difference}=4$

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.

$1, 5, 9, \square, \triangle$

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

$9+4=13\Rightarrow \square=13$ .

The sequence is now,

$1, 5, 9, 13, \triangle$

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.

$13+4=17\Rightarrow \triangle=17$

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Question

What are the next two values in the following sequence?

Answer

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.

$\a_1,a_2,a_3,... \ \ \frac{a_2}{a_1}=d \ \ \frac{a_3}{a_2}=d$

Given this particular sequence

the common ratio using the above method is as follows.

$\a_1,a_2,a_3,... \ \ \frac{7}{1}=7 \ \ \frac{49}{7}=7 \ \textup{Common Difference}=7$

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,

$49\times 7=343$ .

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.

$343\times 7=2401$

Step 4: Answer the question.

The next two terms in the sequence are, .

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Question

What is the correct function that describes the following sequence?

Assume the sequence starts with the input value of zero.

Answer

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,

$\f(0)=3 \f(1)=4 \f(2)=6 \f(3)=10 \f(4)=18 \f(5)=34$

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.

$\f(0)=a^0+b=3 \f(1)=a^1+b=4 \f(2)=a^2+b=6$

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

$\f(0)=a^0+b \f(0)=1+b \1+b=3 \b=2$

Substituting the value for into the function, solve for .

$\f(n)=a^n+2 \f(1)=a^1+2 \4=a^1+2$

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

$\4=a+2 \2=a$

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

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Question

Find the 26th term in the following sequence.

Assume the sequence starts with input of one.

Answer

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,

$\f(1)=0 \f(2)=2 \f(3)=6 \f(4)=14 \f(5)=30$

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.

$\f(3)=2(2-0)+2 \f(3)=4+2 \f(3)=6$

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.

$\f(1)=2^1-2=0 \f(2)=2^2-2=4-2=2 \f(3)=2^3-2=8-2=6 \f(4)=2^4-2=16-2=14 \f(5)=2^5-2=32-2=30$

Step 4: Calculate the specific value in question.

$\ f(26)=2^{26}-2 \f(26)=67108864-2 \f(26)=67108862$

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Question

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

$225,196,169,\triangle, 121, \square$

Answer

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.

$225,196,169,\triangle, 121, \square$

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.

$n=a^2\Rightarrow \sqrt{n}=a$

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

$225,196,169,\triangle, 121, \square$

Becomes,

$15^2,14^2,13^2,\bigtriangleup, 11^2, \square$ .

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

$15^2,14^2,13^2,\bigtriangleup, 11^2, \square\Rightarrow 15^2,14^2,13^2,12^2, 11^2,10^2$

Therefore,

$\ \triangle=12^2=144 \ \square=10^2=100$

Step 4: Answer the question.

$\square=100$

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Question

Which function correctly describes the following sequence.

Assume the sequence starts with the input value of zero.

Answer

his 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,

$\f(0)=4 \f(1)=6 \f(2)=12 \f(3)=30 \f(4)=84$

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.

$\f(0)=a^0+b=4 \f(1)=a^1+b=6 \f(2)=a^2+b=12$

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

$\f(0)=a^0+b \f(0)=1+b \1+b=4 \b=3$

Substituting the value for into the function, solve for .

$\f(n)=a^n+3 \f(1)=a^1+3 \6=a^1+3$

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

$\6=a+3 \3=a$

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

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Question

Given that the Fibonacci sequence is

for all

$n\geq1$ .

What is the sequence for the first eight entries.

Answer

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 $n\geq1$

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,

$\f(1)=1 \f(2)=f(1)+f(0)=1 \f(3)=f(2)+f(1)=1+1=2 \f(4)=f(3)+f(2)=2+1=3 \f(5)=f(4)+f(3)=3+2=5 \f(6)=f(5)+f(4)=5+3=8 \f(7)=f(6)+f(5)=8+5=13 \f(8)=f(7)+f(6)=13+8=21$

Step 3: Answer the question.

The sequence for the first eight entries

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