Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{0.8,-4,20,-100, \ldots\\}\)

Step-by-Step Solution

Answer

The explicit formula is \(a_n = 0.8 (-5)^{n-1}\).

Step 1: Identify the First Term

The first term of the sequence, denoted as \(a_1\), is \(0.8\).

Step 2: Identify the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(-4 \div 0.8 = -5\). Therefore, the common ratio \(r\) is \(-5\).

Step 3: Write the Explicit Formula

The formula for the \(n^{th}\) term of a geometric sequence is given by \(a_n = a_1 \times r^{n-1}\). Substitute \(a_1 = 0.8\) and \(r = -5\) to get: \[a_n = 0.8 \times (-5)^{n-1}\].

Key Concepts

Explicit Formula

The explicit formula for a geometric sequence is a powerful tool. It allows us to find any term in the sequence quickly, without having to compute all previous terms. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the "common ratio." This makes the explicit formula for the sequence look like:

• \(a_n = a_1 \times r^{n-1}\)

where:

• \(a_n\) is the \(n^{th}\) term you want to find,
• \(a_1\) is the first term, and
• \(r\) is the common ratio.

By using this formula, you can easily calculate any term in the sequence by plugging in the first term \(a_1\), the common ratio \(r\), and the position \(n\) of the term you need.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply one term to get the next term. This ratio is key to determining the behavior of the sequence. To find the common ratio \(r\), you divide any term by the preceding term. Let's look at how it's done using the given sequence:

• Second term \(-4\) divided by the first term \(0.8\): \(-4 \div 0.8 = -5\)

Thus, the common ratio \(r\) is \(-5\).
The sign of the common ratio also tells you something important:

• A positive ratio means the sequence terms will all have the same sign.
• A negative ratio like \(-5\) results in the signs alternating between positive and negative.

This information can be essential in understanding the overall pattern and direction of your sequence.

First Term

Recognizing the first term of a geometric sequence is critical because it serves as the foundation for calculating all subsequent terms. The first term, often denoted as \(a_1\), sets the starting point for the sequence.
In our sequence \(\{0.8, -4, 20, -100, \ldots \}\), the first term \(a_1\) is \(0.8\). Understanding this initial term helps in constructing the entire sequence using the explicit formula.

• You start with \(a_1\) and multiply by the common ratio to find the next term.
• Continue this multiplication for as many terms as needed, each time using the results to build upon the sequence.

Remember, the choice of first term not only influences the value of the sequence but also its direction and structure, making it a pivotal part of geometric sequences.