Problem 37 - College Algebra

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 given sequence is the first element. Here, the first term, \( a_1 \), is \( 0.8 \).

Step 2: Determine the common ratio

To find the common ratio \( r \), divide the second term by the first term: \(\frac{-4}{0.8} = -5\). So, the common ratio \( r = -5 \).

Step 3: Write the explicit formula for the geometric sequence

The explicit formula for any geometric sequence can be written as \( a_n = a_1 \, r^{n-1} \). Substituting the known values, we get:\( a_n = 0.8 \, (-5)^{n-1} \).

Key Concepts

Understanding the Explicit Formula

An explicit formula for a geometric sequence allows you to find any term in the sequence without knowing the previous one. It's like having a magic formula! For the given sequence, the explicit formula is calculated using the first term and the common ratio. The general form of an explicit formula for a geometric sequence is given by:

\[a_n = a_1 \, r^{n-1}\]

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

In our exercise, \(a_1\) is 0.8 and \(r\) is -5. So, the explicit formula becomes:

\[a_n = 0.8 \, (-5)^{n-1}\]With this formula, you can easily find any term in this geometric sequence by just plugging in the value of \(n\). This formula saves time and makes calculations straightforward!

Discovering the Common Ratio

The common ratio is a fundamental component of a geometric sequence. It tells us the factor by which we multiply a term to get the next one. In simple terms, it's what makes a geometric sequence special. To find the common ratio \(r\), you simply divide any term in the sequence by its preceding term.

In the provided sequence \(\{0.8, -4, 20, -100, \ldots\}\), \(r\) was calculated by dividing the second term by the first:

\[r = \frac{-4}{0.8} = -5\]

• This means that every term is -5 times the term before it.
• Notice that the sign plays a crucial role; it indicates that the sequence alternates signs with each term.

Understanding the common ratio helps you predict the behavior of the sequence and use it effectively in the explicit formula.

Exploring Sequence Terms

Sequence terms are the individual numbers that make up a sequence. In a geometric sequence, each term is derived from the previous one by multiplying with the common ratio. This gives the sequence its unique pattern.

Let's explore the terms in a geometric sequence:

• The first term \(a_1\) sets the base for the entire sequence; here it is 0.8.
• The second term \(a_2\) is calculated as \(a_1 \times r\), giving us -4.
• Similarly, the third term \(a_3\) can be found by multiplying \(a_2\) with \(r\) again, which equals 20.
• This pattern continues, resulting in a sequence that expands as \{0.8, -4, 20, -100,\ldots\}.

Each term depends on both the previous term and the common ratio, exhibiting an exponential-like growth or decay, which is distinct to geometric sequences.