Problem 33 - College Algebra

Question

For the following exercises, write the first five terms of the geometric sequence. $$ a_{n}=12 \cdot\left(-\frac{1}{2}\right)^{n-1} $$

Step-by-Step Solution

Answer

The first five terms are 12, -6, 3, -1.5, and 0.75.

Step 1: Understanding the Sequence Formula

The formula for a geometric sequence is given as \( a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1} \). Here, 12 is the first term \( a_1 \) of the sequence, and \(-\frac{1}{2}\) is the common ratio, \( r \). The exponent \( n-1 \) indicates the position of the term in the sequence.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \[a_1 = 12 \cdot \left(-\frac{1}{2}\right)^{1-1} = 12 \cdot 1 = 12.\]The first term \( a_1 \) is 12.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \[a_2 = 12 \cdot \left(-\frac{1}{2}\right)^{2-1} = 12 \cdot \left(-\frac{1}{2}\right) = -6.\]The second term \( a_2 \) is -6.

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \[a_3 = 12 \cdot \left(-\frac{1}{2}\right)^{3-1} = 12 \cdot \left(\frac{1}{4}\right) = 3.\]The third term \( a_3 \) is 3.

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \[a_4 = 12 \cdot \left(-\frac{1}{2}\right)^{4-1} = 12 \cdot \left(-\frac{1}{8}\right) = -1.5.\]The fourth term \( a_4 \) is -1.5.

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula: \[a_5 = 12 \cdot \left(-\frac{1}{2}\right)^{5-1} = 12 \cdot \left(\frac{1}{16}\right) = 0.75.\]The fifth term \( a_5 \) is 0.75.

Key Concepts

Understanding the Sequence Formula

In mathematics, a sequence is a list of numbers arranged in a specific order. A geometric sequence is a type of numerical sequence where each term after the first is found by multiplying the previous term by a fixed number called the *common ratio*. The sequence formula is crucial for defining and understanding how these sequences behave.
For this exercise, the geometric sequence is presented by the formula:

• \( a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1} \)

Here, \( a_n \) represents the term in the sequence, \( n \) is the term number, and \( 12 \) is used as the starting point or the first term of the sequence. The formula emphasizes how each term can be calculated based on its position \( n \). Understanding this relationship is key to mastering geometric sequences. The formula essentially provides a blueprint for computing any term within the sequence efficiently.

Identifying the First Term

The first term of a geometric sequence is fundamental as it initiates the entire pattern of the sequence. In our sequence formula:

• \( a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1} \),

The first term, \( a_1 \), is found by substituting \( n = 1 \) into the sequence formula. Substituting gives us:

• \( a_1 = 12 \cdot \left(-\frac{1}{2}\right)^{1-1} = 12 \cdot 1 = 12 \)

Thus, the first term of this sequence is 12. This initial value sets the stage for all subsequent terms, each calculated via multiplication by the common ratio. Recognizing the significance of this term helps to quickly grasp how the progression begins, laying the foundation for all other terms.

Determining the Common Ratio

The *common ratio* in a geometric sequence is the fixed factor by which each term is multiplied to obtain the next term. Identifying this ratio is key to understanding the sequence's behavior. Based on our formula:

• \( a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1} \)

The common ratio \( r \) is \(-\frac{1}{2}\). This means that each subsequent term is half of the previous term, but with alternating signs. To find the common ratio, you can also divide any term by the previous term as a check:

• For example, \( a_2 = -6 \) and \( a_1 = 12\).
• So, \( r = \frac{-6}{12} = -\frac{1}{2}\).

By understanding this multiplication pattern, you can predict how the terms behave as the sequence progresses, and calculate terms efficiently.

Calculating Terms in the Sequence

Term calculation is the process of finding any term in the sequence using the formula provided. Knowing the sequence formula allows you to determine terms without manually computing each one by multiplying repeatedly. With the formula:

• \( a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1} \)

You can find any term by substituting the term number \( n \) into the formula. For instance:

• The second term, \( a_2 \), is calculated as: \[ a_2 = 12 \cdot \left(-\frac{1}{2}\right)^{2-1} = -6 \]
• The third term, \( a_3 \), is calculated as: \[ a_3 = 12 \cdot \left(-\frac{1}{2}\right)^{3-1} = 3 \]
• Continuing this way lets you find any term directly.

This method is quick and prevents errors from manual calculation, making it an efficient approach to working with geometric sequences.