Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, write the first eight terms of the piecewise sequence. \(a_{n}=\left\\{\begin{array}{ll}(2 n+1)^{2} & \text { if } n \text { is divisible by } 4 \\ \frac{2}{n} & \text { if } n \text { is not divisible by } 4\end{array}\right.\)

Step-by-Step Solution

Answer

The first eight terms are: 2, 1, \(\frac{2}{3}\), 81, \(\frac{2}{5}\), \(\frac{1}{3}\), \(\frac{2}{7}\), 289.

Step 1: Understand the Piecewise Function

We need to generate the first eight terms of the sequence given by a piecewise function. The function provides two different expressions depending on the divisibility of \(n\) by 4.

Step 2: Calculate Terms for n Divisible by 4

For values of \(n\) that are divisible by 4 (such as 4 and 8 in our range), we use the expression \((2n+1)^2\).

Step 3: Calculate Terms for n Not Divisible by 4

For values of \(n\) not divisible by 4 (1, 2, 3, 5, 6, 7 in our range), we use the expression \(\frac{2}{n}\).

Step 4: Compute Terms 1 to 4

- For \(n = 1\), since 1 is not divisible by 4, use \(\frac{2}{1} = 2\).- For \(n = 2\), use \(\frac{2}{2} = 1\).- For \(n = 3\), use \(\frac{2}{3}\).- For \(n = 4\), use \((2\cdot4+1)^2 = 81\).

Step 5: Compute Terms 5 to 8

- For \(n = 5\), use \(\frac{2}{5}\).- For \(n = 6\), use \(\frac{2}{6} = \frac{1}{3}\).- For \(n = 7\), use \(\frac{2}{7}\).- For \(n = 8\), use \((2\cdot8+1)^2 = 289\).

Step 6: Write the First Eight Terms

The first eight terms of the sequence are: 2, 1, \(\frac{2}{3}\), 81, \(\frac{2}{5}\), \(\frac{1}{3}\), \(\frac{2}{7}\), 289.

Key Concepts

Sequence Terms

In mathematics, a sequence is an ordered list of numbers that follow a particular pattern or rule. Each value in the sequence is known as a "term." Each term of a sequence can be described using a mathematical expression, which allows us to calculate the term based on its position in the sequence.
In the exercise, the task is to find the first eight sequence terms. These terms are determined by a piecewise function, meaning they arise from different expressions depending on their position.

• The first term corresponds to when the position, denoted by \(n\), is 1.
• The second term corresponds to \(n = 2\), and so on, up to \(n = 8\).

As we fill in the sequence, we determine each term by checking if \(n\) is divisible by 4 or not, and applying the corresponding expression.

Divisibility

Divisibility determines whether one integer can be divided by another integer without leaving a remainder. It is especially important in sequence problems that include expressions conditioned by divisibility.
In the given piecewise sequence, we need to check the divisibility of \(n\) by 4:

• If \(n\) is divisible by 4, it means dividing \(n\) by 4 leaves no remainder. For instance, 4 and 8 are divisible by 4.
• If \(n\) is not divisible by 4, there will be a remainder left when dividing \(n\) by 4. Numbers like 1, 2, 3, 5, 6, and 7 are not divisible by 4 in our range.

Understanding and checking divisibility by 4 guides choosing the correct expression for each sequence term as defined by the piecewise function.

Expression Evaluation

Evaluating an expression means calculating its value based on given conditions and inputs. For a piecewise sequence, we sometimes need to evaluate different expressions to find the specific term.
To evaluate the terms of this sequence, follow:

• For \(n\) divisible by 4: Use the expression \((2n+1)^2\). For example, for \(n = 4\), replace \(n\) with 4 in the expression to get \((2 \cdot 4 + 1)^2 = 81\).
• For \(n\) not divisible by 4: Use \(\frac{2}{n}\). So, when \(n = 2\), substitute \(n\) with 2 in the expression to find \(\frac{2}{2} = 1\).

This step-by-step evaluation is key to finding each term of the piecewise sequence accurately.

Piecewise Expressions

Piecewise functions use different expressions based on certain conditions. A single function consists of multiple "pieces," allowing it to manage different rules under different circumstances.
The current exercise uses a piecewise-defined sequence where:

• One piece, \((2n+1)^2\), is used when \(n\) is divisible by 4.
• The other, \(\frac{2}{n}\), is used when \(n\) is not divisible by 4.

The condition of each "piece" is determined by examining the input's properties, such as divisibility by 4 here. Piecewise expressions are useful for describing complex sequences where different rules apply to different segments of the input.