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

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Example Question #31 : Interpreting Functions

Which function correctly describes the following sequence.

4,6,12,30,84,...

Assume the sequence starts with the input value of zero.

Possible Answers:

$f(n)=3n^{-n}$

$f(n)=2n^{n}-1$

$f(n)=2n^{n}$

f(n)=3^n+3

$f(n)=n^{n-1}$

Correct answer:

f(n)=3^n+3

Explanation:

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,

4,6,12,30,84,...

the following logic statement can be created.

Let f(n) 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 f(0) 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.

f(n)=a^n+b 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.

f(n)=3^n+3

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Example Question #32 : Interpreting Functions

What is the twelfth term in the Fibonacci sequence, (1, 1, 2, 3, 5, 8, ...)?

Possible Answers:

144

233

Correct answer:

144

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,

f(n+1)=f(n)+f(n-1) 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,

(1, 1, 2, 3, 5, 8, ...)

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

Study concepts, example questions & explanations for Common Core: High School - Functions

All Common Core: High School - Functions Resources

Example Questions

Example Question #31 : Interpreting Functions

Example Question #32 : Interpreting Functions

All Common Core: High School - Functions Resources

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