Problem 34 - College Algebra with Corequisite Support

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

Step 1: Identify the first term

In a geometric sequence, the first term is denoted as \(a_1\). For the sequence \( \{-2, -4, -8, -16, \ldots\} \), the first term \(a_1\) is \(-2\).

Step 2: Determine the common ratio

The common ratio \(r\) in a geometric sequence is found by dividing the second term by the first term. For this sequence, \(-4 \div (-2) = 2\). Thus, the common ratio is \(2\).

Step 3: Write the general formula

The explicit formula for a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\). Substitute \(a_1 = -2\) and \(r = 2\) into the formula: \(a_n = -2 \cdot 2^{n-1}\).

Key Concepts

Explicit Formula

In mathematics, an explicit formula is essential for understanding geometric sequences. This formula provides a specific expression to find any term in the sequence using its position. By knowing just a few details about the sequence, it allows you to compute any element without needing to iterate through the entire sequence. Rarely will you have to calculate each step when you use an explicit formula. It's like having a shortcut to reach the desired term directly.

For a geometric sequence, the explicit formula is expressed as:- \( a_n = a_1 \cdot r^{n-1} \)
Here:

• \(a_n\) represents the term you want to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) indicates the position in the sequence.

Using this formula, you can determine any term's value with a straightforward calculation. For instance, to find the 5th term in our sequence from the exercise, substitute the values: \( a_5 = -2 \cdot 2^{5-1} = -2 \cdot 16 = -32 \). The explicit formula is powerful in sequence analysis.

Common Ratio

The common ratio, denoted as \(r\), is a fundamental feature of geometric sequences. It serves as the link between consecutive terms, multiplying each term by the same constant to yield the next. Understanding the common ratio helps us gain insight into the pattern of the sequence and predict future terms easily.

You can identify the common ratio by dividing any term in the sequence by the preceding term:- For instance, in \{-2, -4, -8, -16, \ldots\}, dividing the second term by the first term: \(-4 \div -2 = 2\).- Therefore, the common ratio \(r = 2\).

• A positive \(r\) indicates consistent direction, either all positive or negative.
• A negative \(r\) means alternating signs between terms.
• If \(|r| > 1\), the sequence grows or shrinks exponentially.
• If \(0 < |r| < 1\), the sequence gradually converges towards zero.

Understanding the common ratio makes recognizing and working with geometric sequences smoother.

Geometric Sequence Formula

The geometric sequence formula is your blueprint for understanding how sequences are structured and how to compute various aspects like future terms. It combines the initial term and the common ratio to express the entire sequence.

The formula is written as:- \( a_n = a_1 \cdot r^{n-1} \)
In this expression:

• The sequence commences with \(a_1\), the initial term.
• Each subsequent term is calculated by multiplying the previous term by \(r\), the common ratio.

To apply this formula, you simply need the first term and the common ratio of the sequence. The formula, as derived in the exercise, showcases:- \(a_n = -2 \cdot 2^{n-1} \)This succinct representation can quickly help you estimate any term within the sequence without recursion.

Understanding the geometric sequence formula is crucial for efficient calculations and can be applied in various fields like physics, finance, and data analysis.