Problem 34 - College Algebra

Question

For the following exercises, write an explicit formula for each geometric sequence. $$ a_{n}=\\{-2,-4,-8,-16, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \(a_n = -2 \cdot 2^{n-1}\).

Step 1: Identify Sequence Characteristics

The sequence given is \(-2, -4, -8, -16, \ldots\). We need to determine if it is a geometric sequence. For a geometric sequence, each term is the previous term multiplied by a constant ratio. Let's find this common ratio.

Step 2: Find the Common Ratio

To find the common ratio \(r\), we divide the second term by the first term. So, \(r = \frac{-4}{-2} = 2\). We can verify this with the next terms: \(-8 \div -4 = 2\) and \(-16 \div -8 = 2\). Therefore, the common ratio \(r = 2\).

Step 3: Determine the First Term

The first term of the sequence is \(a_1 = -2\). We'll use this to write the explicit formula for the sequence.

Step 4: Write the Explicit Formula

A geometric sequence has the general form \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. Substitute the values we found: \(a_n = -2 \cdot 2^{n-1}\). This is the explicit formula for the sequence.

Key Concepts

Explicit Formula

An explicit formula is a tool that lets you find any term in a geometric sequence without listing out all the previous terms. For a geometric sequence, the explicit formula is given by:
\[a_n = a_1 \cdot r^{n-1}\]In this formula:

• \(a_n\) represents the term number \(n\) in the sequence.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio, which indicates how each term is obtained by multiplying the previous term.

By using the explicit formula, it becomes easy to calculate any term in the sequence directly. For example, in our solved sequence, we have:

• \(a_1 = -2\)
• \(r = 2\)

Thus, our explicit formula becomes \(a_n = -2 \cdot 2^{n-1}\). This formula allows us to find any term \(n\) by just plugging in the value of \(n\).

Common Ratio

The concept of a common ratio is essential in recognizing and working with geometric sequences. A geometric sequence is characterized by this common ratio, which is a constant factor between consecutive terms. In mathematical terms, for a sequence to be geometric:
The ratio \(r\) is found by dividing any term in the sequence by the previous one:
\[r = \frac{a_{n}}{a_{n-1}}\]Let's take a closer look at our sequence:

• The second term \(-4\) divided by the first term \(-2\) gives \(r = 2\).
• This operation can be repeated for other consecutive terms to confirm: \(\frac{-8}{-4} = 2\) and \(\frac{-16}{-8} = 2\).

Repeating this for each consecutive pair allows us to be certain that \(r = 2\) consistently. This constant value is what defines the geometric nature of the sequence. Knowing the common ratio is crucial for using the explicit formula effectively.

Sequence Characteristics

Understanding the characteristics of a sequence helps to determine if it is geometric and allows the usage of specific mathematical formulas. Here are the steps to identify sequence characteristics:

• Check if the sequence follows a multiplying pattern, as opposed to linear additions or subtractions. Geometric sequences multiply a fixed number (the common ratio) between terms.
• Verify by checking the common ratio for consistency across consecutive terms, ensuring it remains constant.
• Note the first term, \(a_1\), because it is vital for writing the explicit formula.

In our example, the sequence \(-2, -4, -8, -16, \ldots\) is a geometric sequence because each term is multiplied by \(r = 2\) from the previous one. The first term is \(-2\), providing the starting point for our explicit formula. Recognizing these characteristics is the foundation for understanding geometric sequences and predicting future terms effectively.