Problem 15 - College Algebra

Question

Write the first four terms of the sequence. $$a_{n}=-\left(\frac{4 \cdot(-5)^{n-1}}{5}\right)$$

Step-by-Step Solution

Answer

The first four terms are \(-\frac{4}{5}, 4, -20, 100\).

Step 1: Understanding the Sequence Formula

The sequence is given by the formula \( a_n = -\left( \frac{4 \cdot (-5)^{n-1}}{5} \right) \). This formula directs us to plug in sequential integer values starting from 1 for \( n \) to find each term.

Step 2: Finding the First Term

Substitute \( n = 1 \) into the formula: \[a_1 = -\left( \frac{4 \cdot (-5)^{1-1}}{5} \right) = -\left( \frac{4 \cdot 1}{5} \right) = -\frac{4}{5}.\]Thus, the first term \( a_1 \) is \(-\frac{4}{5}\).

Step 3: Finding the Second Term

Substitute \( n = 2 \) into the equation: \[a_2 = -\left( \frac{4 \cdot (-5)^{2-1}}{5} \right) = -\left( \frac{4 \cdot (-5)}{5} \right) = -(-4) = 4.\]So, the second term \( a_2 \) is \(4\).

Step 4: Finding the Third Term

Substitute \( n = 3 \) into the sequence formula: \[a_3 = -\left( \frac{4 \cdot (-5)^{3-1}}{5} \right) = -\left( \frac{4 \cdot 25}{5} \right) = -20.\]The third term \( a_3 \) is \(-20\).

Step 5: Finding the Fourth Term

Substitute \( n = 4 \) into the formula: \[a_4 = -\left( \frac{4 \cdot (-5)^{4-1}}{5} \right) = -\left( \frac{4 \cdot (-125)}{5} \right) = 100.\]Hence, the fourth term \( a_4 \) is \(100\).

Key Concepts

Sequence Formula

A sequence formula is a way of defining a list of numbers in which each number is a term of the sequence. In mathematics, a sequence is an ordered list of numbers and can often be represented by a formula. This formula shows the relationship between each term in the sequence and its position.
In our exercise, the given sequence formula is: \[ a_n = -\left( \frac{4 \cdot (-5)^{n-1}}{5} \right) \]This represents a sequence based on exponents of -5. The formula shows that each term \( a_n \) depends on the position \( n \) and changes as \( n \) increases. To find a specific term, substitute the desired term's position into the formula. The sequence continues as long as integers are substitutable into \( n \).

• Multiplication: The sequence begins with multiplying 4 by a power of -5.
• Exponentiation: The exponent \( n-1 \) alters the power of -5 for each term.
• Division: Finally, the entire product is divided by 5, followed by negation to provide the term.

Understanding each of these components will allow you to calculate any term using the sequence formula.

Finding Terms of a Sequence

Finding terms of a sequence entails using a sequence formula to determine specific numbers at fixed positions in the list. This is usually done by substituting integer values for \( n \), which indicates the position of a term in the sequence.
Here’s how you find each term:
1. **Identify** the sequence formula.2. **Substitute** the integer's position, starting with \( n = 1 \), into the formula.3. **Calculate** step-by-step until the term value is disclosed.
Using our formula, for example:

• For the first term, substitute \( n = 1 \) into the sequence formula:\[ a_1 = -\left( \frac{4 \cdot 1}{5} \right) = -\frac{4}{5} \]
• For the second term, substitute \( n = 2 \):\[ a_2 = -\left( \frac{4 \cdot (-5)}{5} \right) = 4 \]
• Continue this method with \( n = 3 \) and \( n = 4 \) to find subsequent terms.

Repeat the process for as many terms as needed, ensuring that calculations at each step are performed accurately.

Integer Substitution

Integer substitution is a simple yet crucial step in solving sequences where you need to plug whole numbers into a formula to uncover terms. This process forms the basis of evaluating mathematical formulas to find the outcome associated with each sequence position.
To correctly perform integer substitution:

• **Begin with the initial integer** value, usually \( n = 1 \), depending on which term you seek.
• **Replace \( n \) throughout the sequence formula** with this integer. Don’t forget any occurrence of \( n \) in power or division elements.
• Once substituted, **carry out mathematical operations** (like power functions, multiplications, divisions, etc.) in order, keeping mind of the signs.
• **Verify your arithmetic** at each stage to ensure correct computation of each term, especially mindful with exponents and negative numbers.

Through integer substitution, the sequence’s behavior and progression can be explored and understood, allowing learners to predict further terms in the sequence.