Math Problem Statement

Which Function defines the sequence 20, 17, 14, 11 ... and f(2) = -4?

Solution

To determine which function defines the sequence 20,17,14,11,20, 17, 14, 11, \ldots and also satisfies f(2)=4f(2) = -4, let's analyze the given options and the properties of the sequence.

Step 1: Identifying the Sequence Pattern

The sequence provided is 20,17,14,11,20, 17, 14, 11, \ldots. Observing the differences between terms:

  • 1720=317 - 20 = -3
  • 1417=314 - 17 = -3
  • 1114=311 - 14 = -3

The sequence has a common difference of 3-3, which indicates that it is an arithmetic sequence with a slope of 3-3.

Step 2: Applying the Linear Function Form

For an arithmetic sequence, a function describing it can be in the form: f(x)=mx+bf(x) = mx + b where mm is the common difference (slope). Here, m=3m = -3, so we start with: f(x)=3x+bf(x) = -3x + b

Step 3: Finding the Value of bb

Since f(2)=4f(2) = -4, we substitute x=2x = 2 and f(x)=4f(x) = -4 into the function: 3(2)+b=4-3(2) + b = -4 6+b=4-6 + b = -4 b=2b = 2

Thus, the function is: f(x)=3x+2f(x) = -3x + 2

Step 4: Verifying the Answer

Looking at the options, the correct choice is: f(x)=3x+2\boxed{f(x) = -3x + 2}

Would you like further details, or do you have any questions?

Here are some related questions you might explore:

  1. How would the function change if the common difference were different?
  2. Can you use this method to define other types of sequences?
  3. What would the function be if f(2)=5f(2) = 5 instead?
  4. How do you determine the common difference in sequences?
  5. What would the function look like if the sequence were geometric?

Tip: For arithmetic sequences, finding the common difference is essential to defining the sequence with a linear function.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequence
Linear Functions

Formulas

f(x) = mx + b

Theorems

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Suitable Grade Level

Grades 7-9