Problem 59 - Precalculus with Limits

Question

In Exercises 59 - 64, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so,find the model. \( 5, 13, 21, 29, 37, 45, \cdots \)

Step-by-Step Solution

Answer

Yes, the sequence can be perfectly represented by a linear model and that model is \( f(x) = 8x + 5\)

Step 1: Identify the Sequence Type

A Linear arithmetic sequence will have a common difference whereas a Quadratic sequence will have a common second difference. In this case, the differences of the consecutive terms of the sequence are: 13-5=8, 21-13=8, 29-21=8, 37-29=8, 45-37=8. Thus, the common difference is 8. The sequence is linear.

Step 2: Find the Linear Equation

A linear sequence can be represented as a linear function in the form f(x) = mx + c, where m represents the common difference and c is the initial term. Here, m is 8 and the initial term c is 5. So the linear function which perfectly fits the sequence is f(x) = 8x + 5.

Step 3: Test the Model

Confirm that the generated model correctly represents the sequence by substituting some x values in the function and check whether it produces the sequence. Let's confirm with first term and second term. Substitute x = 1 in f(x), f(1) = 8*1 + 5= 13 and for second term, f(2) = 8*2 + 5 = 21. Both results match the sequence, therefore, the linear model is a good fit for the sequence.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the "common difference." In the sequence given, 5, 13, 21, 29, 37, 45, ..., we see that each term is increased by a consistent number from the previous one. To check if a sequence is arithmetic, just subtract one term from the next in the sequence:

• From 5 to 13, the difference is 8.
• From 13 to 21, the difference is also 8.
• Continuing this process, we find that the difference remains constant at 8, confirming it's arithmetic.

This constant nature of addition, the common difference, makes predicting future terms straightforward, making arithmetic sequences incredibly useful in various mathematical applications.

Common Difference

The common difference is a critical component of an arithmetic sequence and is what distinguishes it from other types of sequences. In our given sequence, the common difference is 8, as each term increases by this amount. The calculation is simple:

• Take the second term and subtract the first: 13 - 5 = 8.
• Do the same for any other pair of consecutive terms, and you’ll find they all give the same result: 21 - 13 = 8, and so on.

Identifying this common difference allows us to write down a formula for the sequence, as it tells us precisely "how much" to add to any term to get the next one. More broadly, this concept is essential in various branches of mathematics, especially when dealing with linear phenomena.

Linear Function

A linear function is a mathematical equation that models a straight line and is expressed in the form of \( f(x) = mx + c \), where \( m \) represents the "slope" or "rate of change," and \( c \) is the "y-intercept." For an arithmetic sequence, the slope corresponds to the common difference, and the y-intercept is the first term of the sequence. In our case:

• The common difference \( m \) is 8, as we've calculated from the sequence.
• The first term \( c \), which is the starting point of our function, is 5.

Thus, the sequence can be represented by the linear function \( f(x) = 8x + 5 \). This function allows us to determine any term in the sequence by substituting \( x \) with the position of the term we want to find. This is a powerful tool because it provides a quick and easy way to find any number in the sequence, without needing to sum up all previous terms.