Problem 18 - College Algebra

Question

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

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: Determine if n is divisible by 4

To find the first eight terms, we need to check for each \(n\) from 1 to 8 whether \(n\) is divisible by 4. A number is divisible by 4 if it can be evenly divided by 4 without a remainder. Thus, check \(n = 4\) and \(n = 8\) as they are divisible by 4.

Step 2: Calculate terms for n divisible by 4

For \(n\) divisible by 4, use the rule \((2n+1)^2\). For \(n=4\), calculate \((2 \times 4 + 1)^2 = 9^2 = 81\). For \(n=8\), calculate \((2 \times 8 + 1)^2 = 17^2 = 289\). Thus, terms for \(n = 4\) and \(n = 8\) are 81 and 289, respectively.

Step 3: Calculate terms for n not divisible by 4

For \(n\) not divisible by 4, use the rule \(\frac{2}{n}\). Calculate for \(n=1\), \(n=2\), \(n=3\), \(n=5\), \(n=6\), \(n=7\). So, we get: \(\frac{2}{1}=2\), \(\frac{2}{2}=1\), \(\frac{2}{3}=\frac{2}{3}\), \(\frac{2}{5}=\frac{2}{5}\), \(\frac{2}{6}=\frac{1}{3}\), \(\frac{2}{7}=\frac{2}{7}\).

Step 4: Compile the complete sequence

Based on the calculations: From Step 2, \(a_4 = 81\) and \(a_8 = 289\). From Step 3, \(a_1 = 2\), \(a_2 = 1\), \(a_3 = \frac{2}{3}\), \(a_5 = \frac{2}{5}\), \(a_6 = \frac{1}{3}\), \(a_7 = \frac{2}{7}\). Thus, the sequence is: 2, 1, \(\frac{2}{3}\), 81, \(\frac{2}{5}\), \(\frac{1}{3}\), \(\frac{2}{7}\), 289.

Key Concepts

Divisibility by 4

Understanding whether a number is divisible by 4 is key to solving problems involving piecewise sequences. A number is divisible by 4 if, when divided by 4, there is no remainder.
This means the number fits perfectly within the multiples of 4: 4, 8, 12, and so on.
So, to identify these numbers:

• Imagine dividing the number by 4.
• If the result is a whole number, it is divisible by 4.

In our exercise, we focus on the numbers from 1 to 8.
Only 4 and 8 meet this criterion since both divide evenly by 4.
By correctly identifying these numbers, the appropriate terms of the sequence can be calculated.

Sequence Terms

Piecewise sequences are defined with different rules for different values of their index, often based on a condition.
This exercise features a sequence with a rule specified by divisibility:

• If the index \( n \) is divisible by 4, the term is determined using the formula \((2n+1)^2\).
• When 4 doesn't divide \( n \) evenly, the term is calculated as \( \frac{2}{n} \).

Piecewise sequences provide valuable versatility, enabling distinct formulas based on conditions like divisibility.
This results in sequences that can behave quite differently depending on the conditions outlined.

Calculation Steps

Piecewise sequences require calculating terms step by step, respecting each condition.
In this case, execute by following the sequence rules, depending on the divisibility criterion:- **Divisible by 4:** - For \( n = 4 \): Use the formula \( (2n+1)^2 \) to get \((2 \times 4 + 1)^2 = 81\). - For \( n = 8 \): Again using \((2n+1)^2\), calculate \((2 \times 8 + 1)^2 = 289\). - Thus, the terms are 81 and 289.- **Not Divisible by 4:** - For the remaining numbers (1, 2, 3, 5, 6, 7), apply \( \frac{2}{n} \) to find: - \( a_1 = \frac{2}{1} = 2 \) - \( a_2 = \frac{2}{2} = 1 \) - \( a_3 = \frac{2}{3} \) - \( a_5 = \frac{2}{5} \) - \( a_6 = \frac{1}{3} \) - \( a_7 = \frac{2}{7} \)By following these detailed steps, we've constructed the sequence terms individually, resulting in a complete sequence.