Problem 38 - College Algebra

Question

For the following exercises, write an explicit formula for each geometric sequence. $$ a_{n}=\\{-1.25,-5,-20,-80, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = -1.25 \, (4)^{n-1} \).

Step 1: Identify the First Term

The first term of the geometric sequence, denoted as \( a_1 \), is the initial term of the list provided. Here, \( a_1 = -1.25 \).

Step 2: Determine the Common Ratio

To find the common ratio \( r \) of the geometric sequence, divide the second term by the first term. Using the sequence: \(-5 \div -1.25 = 4\). Similarly, check using subsequent terms: \(-20 \div -5 = 4\) and \(-80 \div -20 = 4\). Thus, the common ratio \( r = 4 \).

Step 3: Write the Explicit Formula

The general explicit formula for a geometric sequence is \( a_n = a_1 \, r^{(n-1)} \). Substitute the known values \( a_1 = -1.25 \) and \( r = 4 \) to get: \[ a_n = -1.25 \, (4)^{(n-1)} \].

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression used to calculate any term in a sequence directly, without needing to find the previous terms. In the context of a geometric sequence, the explicit formula provides a handy way to find the nth term by using the first term and the common ratio. This avoids the need for step-by-step calculations through the entire sequence.The general form of the explicit formula for a geometric sequence is given by:\[ a_n = a_1 \cdot r^{(n-1)} \]where:

• \( a_n \) is the nth term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

This formula is very powerful because it uses only the first term and the common ratio to find any term in the sequence. Substitute the specific values for \( a_1 \) and \( r \) to get the formula for your particular sequence.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply one term to get the next term. Identifying this constant multiplier is crucial since it helps establish the relationship between consecutive terms of the sequence.To determine the common ratio, take any term in the sequence (apart from the first) and divide it by the term just before it. In our exercise, we found the common ratio by performing the following calculations:

• \( -5 \div -1.25 = 4 \)
• \( -20 \div -5 = 4 \)
• \( -80 \div -20 = 4 \)

As seen, each division confirms that the common ratio \( r = 4 \). This tells us that to get from one term to the next, we multiply by 4. A consistent common ratio is a defining characteristic of geometric sequences.

First Term

The first term in a geometric sequence is the starting point for creating the sequence. It's represented as \( a_1 \) and serves as the base value that will be multiplied repeatedly by the common ratio to create the sequence.Identifying the first term is straightforward. It's simply the first number in the sequence list. In our case, the sequence \(-1.25, -5, -20, -80, \ldots\) begins with \( a_1 = -1.25 \).Understanding the first term is crucial because it lays the foundation for the entire sequence. Without the first term, we cannot apply the explicit formula to find other terms in the sequence. It joins forces with the common ratio to determine the behavior and pattern of the sequence. Remember, in the explicit formula, the first term is multiplied by the common ratio raised to powers increasing with \( n \). Thus, knowing \( a_1 \) is always your first step.