Problem 24 - Precalculus Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is geometric. If it is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1}\). $$a_{n}=(-1)^{n} 2^{n}$$

Step-by-Step Solution

Answer

The first five terms are -2, 4, -8, 16, -32. It is a geometric sequence with a common ratio of -2.

Step 1: Calculate the First Term

To find the first term of the sequence, substitute \( n = 1 \) into the expression \( a_n = (-1)^n 2^n \). \[ a_1 = (-1)^1 2^1 = -2 \].

Step 2: Calculate the Second Term

To find the second term, substitute \( n = 2 \) into the expression. \[ a_2 = (-1)^2 2^2 = 4 \].

Step 3: Calculate the Third Term

To find the third term, substitute \( n = 3 \) into the expression. \[ a_3 = (-1)^3 2^3 = -8 \].

Step 4: Calculate the Fourth Term

To find the fourth term, substitute \( n = 4 \) into the expression. \[ a_4 = (-1)^4 2^4 = 16 \].

Step 5: Calculate the Fifth Term

To find the fifth term, substitute \( n = 5 \) into the expression. \[ a_5 = (-1)^5 2^5 = -32 \].

Step 6: Check if the Sequence is Geometric

A sequence is geometric if the ratio between consecutive terms is constant. Calculate the ratio \( r \) as \( \frac{a_{n+1}}{a_n} \). Check the ratios: \[- \frac{a_2}{a_1} = \frac{4}{-2} = -2 \], \[- \frac{a_3}{a_2} = \frac{-8}{4} = -2 \], \[- \frac{a_4}{a_3} = \frac{16}{-8} = -2 \], and \[- \frac{a_5}{a_4} = \frac{-32}{16} = -2 \]. Since all ratios are equal, the sequence is geometric.

Step 7: Express the nth Term

The nth term of a geometric sequence is expressed as \( a_n = a r^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio. For this sequence, \( a = -2 \) and \( r = -2 \). Therefore, \( a_n = -2(-2)^{n-1} \).

Key Concepts

Understanding the Common Ratio

The common ratio in a geometric sequence is a key feature that defines the nature of the sequence. To determine if a sequence is geometric, we need to check if the ratio between each consecutive term is constant. This constant value is known as the "common ratio."

For any geometric sequence, the common ratio can be calculated using the formula:

• \( r = \frac{a_{n+1}}{a_n} \)

Here, \( a_{n+1} \) represents the next term and \( a_n \) represents the current term. In our worked example, we saw that each ratio indeed turned out to be \(-2\). This uniformity across the series confirms that the sequence is geometric with a common ratio of \(-2\).

Knowing the common ratio is crucial as it helps in formulating the general term and predicting any term's value in the series without listing all prior terms.

Exploring the nth Term

The nth term in a geometric sequence allows us to calculate any term in the sequence by directly using its position. Knowing how to find the nth term, \( a_n \), is valuable as it saves time and effort compared to computing each term sequentially.

The formula for the nth term of a geometric sequence is:

• \( a_n = a \cdot r^{n-1} \)

where:

• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

In the given exercise, we determined that \( a = -2 \) and the common ratio \( r = -2 \). Therefore, the general expression for the nth term is:
\( a_n = -2 \cdot (-2)^{n-1} \).

With this formula, you can find any term in the sequence by plugging in the desired position number into \( n \). This functionality is especially useful for large sequences where calculating all terms is impractical.

Geometric Sequence Formula

A core aspect of working with geometric sequences is understanding and utilizing the geometric sequence formula, which ties together the initial term, the common ratio, and the position in the sequence.

The geometric sequence formula is expressed as:

• \( a_n = a \cdot r^{n-1} \)

Let’s break this down:

• \( a \) is known as the first term and lays the foundation for the sequence.
• \( r \) is the common ratio, which consistently scales each previous term to reach the next one.
• \( n-1 \) accounts for the exponential growth by adjusting the number of multiplications of the common ratio needed.

Using our example, since the first term \( a = -2 \) and the common ratio \( r = -2 \), we plugged these values into the formula to get:
\( a_n = -2 \cdot (-2)^{n-1} \).

This equation empowers us to define the behavior of the sequence entirely based on its core components, making it highly predictable and manageable for mathematical applications.