Determine The Explicit Form Of The Function Represented By The Sequence Below: 3, 7, 11, 15A. F ( N ) = 3 N − 4 F(n)=3n-4 F ( N ) = 3 N − 4 B. F ( N ) = N + 4 F(n)=n+4 F ( N ) = N + 4 C. F ( N ) = 4 N + 3 F(n)=4n+3 F ( N ) = 4 N + 3 D. F ( N ) = 4 N − 1 F(n)=4n-1 F ( N ) = 4 N − 1

Introduction

In mathematics, a sequence is a list of numbers in a specific order. The sequence can be represented by a function, which is a rule that assigns a value to each input. In this article, we will determine the explicit form of the function represented by the sequence 3, 7, 11, 15. We will analyze each option and provide a step-by-step solution to find the correct function.

Understanding the Sequence

The given sequence is 3, 7, 11, 15. To determine the explicit form of the function, we need to find a pattern or rule that generates these numbers. Let's examine the sequence closely:

• The first term is 3.
• The second term is 7, which is 4 more than the first term.
• The third term is 11, which is 4 more than the second term.
• The fourth term is 15, which is 4 more than the third term.

Analyzing the Options

We have four options to choose from:

A. $f(n)=3n-4$ B. $f(n)=n+4$ C. $f(n)=4n+3$ D. $f(n)=4n-1$

Let's analyze each option and determine if it matches the given sequence.

Option A: $f(n)=3n-4$

To test this option, we need to substitute the values of n into the function and see if we get the corresponding terms of the sequence.

• For n = 1, $f(1)=3(1)-4=-1$, which is not equal to 3.
• For n = 2, $f(2)=3(2)-4=2$, which is not equal to 7.
• For n = 3, $f(3)=3(3)-4=5$, which is not equal to 11.
• For n = 4, $f(4)=3(4)-4=8$, which is not equal to 15.

Option A does not match the given sequence.

Option B: $f(n)=n+4$

Let's test this option by substituting the values of n into the function.

• For n = 1, $f(1)=1+4=5$, which is not equal to 3.
• For n = 2, $f(2)=2+4=6$, which is not equal to 7.
• For n = 3, $f(3)=3+4=7$, which is equal to 11.
• For n = 4, $f(4)=4+4=8$, which is not equal to 15.

Option B does not match the given sequence.

Option C: $f(n)=4n+3$

Let's test this option by substituting the values of n into the function.

• For n = 1, $f(1)=4(1)+3=7$, which is equal to 3.
• For n = 2, $f(2)=4(2)+3=11$, which is equal to 7.
• For n = 3, $f(3)=4(3)+3=15$, which is equal to 11.
• For n = 4, $f(4)=4(4)+3=19$, which is not equal to 15.

Option C does not match the given sequence.

Option D: $f(n)=4n-1$

Let's test this option by substituting the values of n into the function.

• For n = 1, $f(1)=4(1)-1=3$, which is equal to 3.
• For n = 2, $f(2)=4(2)-1=7$, which is equal to 7.
• For n = 3, $f(3)=4(3)-1=11$, which is equal to 11.
• For n = 4, $f(4)=4(4)-1=15$, which is equal to 15.

Option D matches the given sequence.

Conclusion

Based on our analysis, the explicit form of the function represented by the sequence 3, 7, 11, 15 is $f(n)=4n-1$. This function generates the sequence by multiplying the input n by 4 and subtracting 1.

Final Answer

The final answer is D. $f(n)=4n-1$.

Introduction

In our previous article, we determined the explicit form of the function represented by the sequence 3, 7, 11, 15. We analyzed each option and found that the correct function is $f(n)=4n-1$. In this article, we will answer some frequently asked questions related to determining the explicit form of a function represented by a sequence.

Q: What is the first step in determining the explicit form of a function represented by a sequence?

A: The first step is to examine the sequence closely and look for a pattern or rule that generates the numbers. This can be done by calculating the differences between consecutive terms or by looking for a common ratio.

Q: How do I determine if a sequence is arithmetic or geometric?

A: An arithmetic sequence has a common difference between consecutive terms, while a geometric sequence has a common ratio between consecutive terms. To determine if a sequence is arithmetic or geometric, calculate the differences or ratios between consecutive terms and look for a pattern.

Q: What is the difference between an explicit and implicit function?

A: An explicit function is a function that can be written in the form $f(x)=y$, where $x$ and $y$ are variables. An implicit function is a function that cannot be written in this form, but can be represented by an equation involving $x$ and $y$. In the context of sequences, explicit functions are typically used to represent the sequence.

Q: How do I determine if a function is linear or non-linear?

A: A linear function is a function that can be written in the form $f(x)=mx+b$, where $m$ and $b$ are constants. A non-linear function is a function that cannot be written in this form. To determine if a function is linear or non-linear, look for a pattern in the sequence or use algebraic techniques to simplify the function.

Q: What is the significance of the explicit form of a function?

A: The explicit form of a function is significant because it provides a clear and concise way to represent the function. This can be useful for a variety of applications, including modeling real-world phenomena, solving equations, and making predictions.

Q: How do I use the explicit form of a function to make predictions?

A: To make predictions using the explicit form of a function, substitute the input values into the function and calculate the corresponding output values. This can be done using algebraic techniques or by using a calculator or computer.

Q: What are some common mistakes to avoid when determining the explicit form of a function?

A: Some common mistakes to avoid include:

• Not examining the sequence closely enough to identify a pattern or rule.
• Not using algebraic techniques to simplify the function.
• Not checking the function for consistency with the sequence.
• Not using a calculator or computer to check the function.

Conclusion

Determining the explicit form of a function represented by a sequence can be a challenging task, but by following the steps outlined in this article and avoiding common mistakes, you can successfully determine the explicit form of a function. Remember to examine the sequence closely, use algebraic techniques to simplify the function, and check the function for consistency with the sequence.

Final Answer

The final answer is that determining the explicit form of a function represented by a sequence requires careful examination of the sequence, use of algebraic techniques to simplify the function, and checking the function for consistency with the sequence.