Problem 11 - College Algebra

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: Find the First Term

To find the first term \(a_1\) of the sequence, substitute \(n = 1\) into the formula \(a_n = 1.25 \cdot (-4)^{n-1}\). This gives \(a_1 = 1.25 \cdot (-4)^{0} = 1.25 \cdot 1 = 1.25\).

Step 2: Calculate the Second Term

Substitute \(n = 2\) into the sequence formula. This results in \(a_2 = 1.25 \cdot (-4)^{1} = 1.25 \cdot (-4) = -5\).

Step 3: Determine the Third Term

Substitute \(n = 3\) into the formula to find the third term. Thus, \(a_3 = 1.25 \cdot (-4)^{2} = 1.25 \cdot 16 = 20\).

Step 4: Compute the Fourth Term

Finally, substitute \(n = 4\) into the formula to obtain the fourth term. Therefore, \(a_4 = 1.25 \cdot (-4)^{3} = 1.25 \cdot (-64) = -80\).

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that allows you to find any term in a sequence without listing all terms before it. The formula typically involves a pattern or a rule that is consistent throughout the sequence.
In the given exercise, the sequence formula is \( a_{n}=1.25 \cdot(-4)^{n-1} \). It defines how each term of the sequence is calculated.The sequence formula often includes:

• An initial term or constant (in this case, 1.25).
• A base number that gets raised to a power (here, \(-4\)).
• A variable that represents the term's position in the sequence \(n\). This position number changes to find each different term.

This formula helps you easily determine any term of the sequence by simply substituting \(n\) with the term's desired position.

Exponential Function

When dealing with sequences that involve powers, you'll often encounter an exponential function. An exponential function is characterized by a constant base raised to a variable exponent.
In this specific arithmetic sequence, the term \((-4)^{n-1}\) represents the exponential part of the function. This component significantly affects the pattern of the sequence because it alters the sign and magnitude of each term.Here's how the exponential function works in this context:

• The base \(-4\) alternates the sign of each term due to its negative value.
• The exponent \(n-1\) increases with each successive term, dictating the multiplication factor.
• The raising of \(-4\) to subsequent powers results in a rapid increase or decrease in the term values.

Understanding how this part operates is crucial for predicting the behavior of the sequence and calculating its terms.

Term Calculation

Term calculation involves applying the sequence formula to find the specific value for each term. Each step in finding a term involves simple substitution and arithmetic operations.
The task is to substitute the term number into the formula and carry out the operations to get the term value.For example, in the sequence \( a_{n}=1.25 \cdot(-4)^{n-1} \):

• For the first term (\(a_1\)), substitute \(n = 1\). Thus, \( a_1 = 1.25 \cdot (-4)^0 = 1.25 \).
• For the second term (\(a_2\)), substitute \(n = 2\). Therefore, \( a_2 = 1.25 \cdot (-4)^1 = -5 \).
• Continue this process for each subsequent term as needed.

Each calculation step provides insight into how patterns and rules embedded in the sequence formula affect the outcomes of each term. By mastering this process, anyone can quickly generate terms, which is particularly useful for sequences with complex alterations.