Problem 19 - College Algebra

Question

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} -0.6 \cdot 5^{n-1} \text { if } n \text { is prime or } 1 \\ 2.5 \cdot(-2)^{n-1} \text { if } n \text { is composite } \end{array}\right. $$

Step-by-Step Solution

Answer

The first eight terms are: -0.6, -3.0, -15.0, -20.0, -75.0, -80.0, -375.0, -320.0.

Step 1: Identify Prime and Composite Numbers

To solve this piecewise sequence, we need to determine whether the numbers from 1 to 8 are prime or composite. Recall: a prime number is greater than 1 and divisible only by itself and 1, while composite numbers have divisors other than 1 or itself. For our purposes here, the numbers from 1 to 8 can be categorized as: - 1 is neither prime nor composite, but per the sequence definition, we treat it like a prime number. - 2, 3, 5, 7 are prime numbers. - 4, 6, 8 are composite numbers.

Step 2: Calculate Terms for Prime or 1

For numbers that are prime or 1, use the formula \[-0.6 imes 5^{n-1}\].1. For \(n=1\), calculate: \(-0.6 imes 5^{0} = -0.6\).2. For \(n=2\), calculate: \(-0.6 imes 5^{1} = -3.0\).3. For \(n=3\), calculate: \(-0.6 imes 5^{2} = -15.0\).4. For \(n=5\), calculate: \(-0.6 imes 5^{4} = -75.0\).5. For \(n=7\), calculate: \(-0.6 imes 5^{6} = -375.0\).

Step 3: Calculate Terms for Composite Numbers

For numbers that are composite, use the formula \[2.5 imes (-2)^{n-1}\].1. For \(n=4\), calculate: \(2.5 imes (-2)^{3} = 2.5 imes (-8) = -20.0\).2. For \(n=6\), calculate: \(2.5 imes (-2)^{5} = 2.5 imes (-32) = -80.0\).3. For \(n=8\), calculate: \(2.5 imes (-2)^{7} = 2.5 imes (-128) = -320.0\).

Step 4: Compile the Sequence

Now that we have calculated the terms, compile them into the sequence: a_1 = -0.6, a_2 = -3.0, a_3 = -15.0, a_4 = -20.0, a_5 = -75.0, a_6 = -80.0, a_7 = -375.0, a_8 = -320.0.

Key Concepts

Prime Numbers

Prime numbers are the building blocks of mathematics. They are natural numbers greater than 1 that cannot be formed by multiplying two smaller natural numbers. This means a prime number is only divisible by 1 and itself, which makes it stand out compared to other numbers. For instance, considering the numbers from 1 to 8, we identify 2, 3, 5, and 7 as prime numbers.

• 2 is the smallest prime number, and it is also the only even prime number.
• 3, 5, and 7 are odd, a common characteristic of prime numbers.

Understanding primes is crucial in number theory and has applications in cryptography, computer algorithms, and more.

Composite Numbers

Composite numbers, as the name suggests, are composed by multiplying smaller numbers. They are natural numbers greater than 1 that have more than two distinct positive divisors. This means composites can be divided evenly by numbers other than 1 and themselves. Observing our set, 4, 6, and 8 are composite numbers.

• 4 is formed by multiplying 2 by 2.
• 6 is the product of 2 and 3.
• 8 results from 2 multiplied by 2 again.

The recognition of composite numbers is fundamental when performing tasks like factoring, simplifying fractions, and solving equations.

Sequence Calculation

In sequence calculation, each term is determined by its position and a specific rule. The sequence given in this exercise is piecewise, meaning it uses different formulas based on number categorization.

For prime numbers or 1, the formula is \(-0.6 \times 5^{n-1}\). This formula requires you to raise 5 to the power of \((n-1)\), then multiply by -0.6. Here's an example:

• For \(n = 2\): Calculate \(-0.6 \times 5^{1} = -3.0\).

For composite numbers, the formula changes to \(2.5 \times (-2)^{n-1}\). You raise -2 to the power of \((n-1)\), then multiply by 2.5:

• For \(n = 4\): Calculate \(2.5 \times (-2)^{3} = -20.0\).

The switch between formulas highlights how the categorization of numbers affects sequence outcomes.

Number Categorization

Number categorization involves classifying numbers into different groups based on their properties. In mathematics, this is crucial for applying appropriate formulas or methods for calculations. In the context of the exercise, categorizing numbers into prime or composite determines which sequence formula to apply.

Key takeaways include:

• Prime categorization involves numbers greater than 1 that aren't divisible by others except 1 and themselves.
• Composite categorization recognizes numbers formed by products of smaller numbers.
• The number 1 is a special case, treated like a prime in this context, even though traditionally it's neither prime nor composite.

Correct categorization leads to accurate sequence term calculation, which is essential in achieving the correct complete sequence.