Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=1.25 \cdot(-4)^{n-1} $$

Step-by-Step Solution

Answer

The first four terms are: 1.25, -5, 20, -80.

Step 1: Identify the Formula

The formula for the sequence is given by \( a_{n} = 1.25 \, \cdot (-4)^{n-1} \). Here, \( 1.25 \) is the coefficient and \( (-4)^{n-1} \) is the base raised to the power \( n-1 \).

Step 2: Calculate the First Term (\(n=1\))

Substitute \( n=1 \) into the sequence formula: \( a_{1} = 1.25 \, \cdot (-4)^{1-1} = 1.25 \, \cdot (-4)^{0} = 1.25 \, \cdot 1 = 1.25 \).

Step 3: Calculate the Second Term (\(n=2\))

Substitute \( n=2 \) into the sequence formula: \( a_{2} = 1.25 \, \cdot (-4)^{2-1} = 1.25 \, \cdot (-4)^{1} = 1.25 \, \cdot (-4) = -5 \).

Step 4: Calculate the Third Term (\(n=3\))

Substitute \( n=3 \) into the sequence formula: \( a_{3} = 1.25 \, \cdot (-4)^{3-1} = 1.25 \, \cdot (-4)^{2} = 1.25 \, \cdot 16 = 20 \).

Step 5: Calculate the Fourth Term (\(n=4\))

Substitute \( n=4 \) into the sequence formula: \( a_{4} = 1.25 \, \cdot (-4)^{4-1} = 1.25 \, \cdot (-4)^{3} = 1.25 \, \cdot (-64) = -80 \).

Key Concepts

First Terms of a Sequence

Understanding the initial terms of a sequence is a fundamental concept in mathematics. Let's delve into how the first few terms of a sequence can be determined. Sequences are essentially ordered lists of numbers that follow a specific pattern or rule. In any sequence, the first terms are like the stepping stones to unlocking the sequence's pattern.

Let's analyze the example with the given formula:

• The first term, often denoted as \(a_1\), is found by substituting the number 1 for \(n\) in the sequence formula. This gives us the starting point of the sequence.
• Similarly, the second term \(a_2\) is found by using \(n=2\), the third term \(a_3\) by \(n=3\), and so on.

Each term builds on the one before it using the sequence’s rule which, in this example, included a consistent mathematical operation applied to each subsequent term.

Sequence Formula

The sequence formula is like a recipe that tells us how to generate each term in a sequence. This formula allows us to predict or construct any term within the sequence without having to list all preceding terms. In this example, the sequence formula is given by:

• \(a_{n} = 1.25 \cdot (-4)^{n-1}\).

Key components of this formula include:

• **Coefficient (1.25):** This number is multiplied by the power of the base. It scales the output of the sequence, altering each term by a consistent multiple.
• **Base (-4):** This is the number raised to the power \(n-1\) in the formula, generating progression in the sequence.
• **Exponent (n-1):** This determines the degree of the base's power and controls the pace at which the sequence's terms grow or shrink.

By using this formula, you can quickly determine any term of the sequence by simply plugging in the value of \(n\).

Power Calculation

Power calculation in sequences involves raising a number to a particular exponent. For this example, the base \(-4\) is raised to the power \(n-1\). Calculating powers is crucial for understanding sequences that involve exponential growth or decay.

Here's how power calculations work:

• To calculate \((-4)^0\), which equals 1. This simplifies calculations dramatically due to any non-zero number raised to the power of 0 equaling 1.
• To calculate \((-4)^1\), which remains as \(-4\). It demonstrates the initial change in sign seen in the sequence.
• To find \((-4)^2\), compute it as 16. Notice how negative numbers raised to even powers become positive.
• To determine \((-4)^3\), calculate it as \(-64\), reflecting how odd powers result in negative outcomes again.

Understanding these power calculations provides insight into how the sequence changes from one term to the next and why alternating patterns might occur.

Arithmetic Operations

Arithmetic operations are fundamental to sequence calculations and involve the basic processes of addition, subtraction, multiplication, and division. In the given sequence, we work primarily with multiplication due to the formula structure.

• Each term involves multiplying the coefficient 1.25 with the calculated power of \(-4\). This operation is straightforward but pivotal, as it affects the outcome of each term by scaling the calculated power.
• Once the power is calculated, the multiplication operation is performed. For instance, in the term calculation \(1.25 \cdot (-4)^n\), once \(-4\) has been powered, it is then multiplied by 1.25.

This operation illustrates how arithmetic manipulations translate numerical operations into meaningful sequence values. Emphasizing these steps helps delineate how arithmetic reasoning is applied consistently across sequences.