Problem 19 - College Algebra

Question

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio. The first term is 16 and the common ratio is \(-\frac{1}{3}\). Find the \(4^{\text {th }}\) term.

Step-by-Step Solution

Answer

The 4th term is \(-\frac{16}{27}\).

Step 1: Identifying the Formula

The formula for the nth term of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you wish to find.

Step 2: Plugging in Known Values

Given that the first term \( a_1 = 16 \), the common ratio \( r = -\frac{1}{3} \), and we want to find the 4th term, we substitute these values into the formula: \( a_4 = 16 \cdot \left(-\frac{1}{3}\right)^{3} \).

Step 3: Calculating the Power of the Common Ratio

Calculate \( \left(-\frac{1}{3}\right)^{3} \). Raising \(-\frac{1}{3}\) to the power of 3 means multiplying it by itself three times: \(-\frac{1}{3} \times -\frac{1}{3} \times -\frac{1}{3} = -\frac{1}{27} \).

Step 4: Computing the 4th Term

Substitute \(-\frac{1}{27}\) back into the equation to find the 4th term: \( a_4 = 16 \times -\frac{1}{27} = -\frac{16}{27} \). Therefore, the 4th term of the sequence is \(-\frac{16}{27}\).

Key Concepts

Common Ratio

In a geometric sequence, the term "common ratio" refers to a fixed number that each term in the sequence is multiplied by to produce the next term. It is a crucial element that defines the sequence's progress from one term to the next.

For instance, if the common ratio is given as \(-\frac{1}{3}\) and the first term is 16, you multiply 16 by \(-\frac{1}{3}\) to get the second term. Then, multiply that result again by \(-\frac{1}{3}\) to get the third term, and so on.

• The common ratio can be positive or negative, which affects the direction and nature of the sequence, with a negative ratio causing the terms to alternate between positive and negative.
• A common ratio of greater than 1 increases the value of the terms, while a common ratio between 0 and 1 decreases them.
• The concept of common ratio is fundamental for understanding how sequences extend and behave.

Nth Term Formula

To find any term in a geometric sequence, you can use the Nth Term Formula, which is expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] Here, \( a_n \) stands for the Nth term you're looking to find, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence.

This formula is crucial for calculating specific terms without having to list all previous terms in the sequence, making it highly efficient.

• Identify your variables: Know your first term, common ratio, and which term you need to find.
• Substitute these values into the formula to perform the calculation.
• This formula showcases the power of exponential growth or decay within geometric sequences.

Understanding this formula is key to predicting the behavior of the sequence at any point.

Sequence Calculation

Calculating terms in a geometric sequence involves not only understanding the common ratio and the Nth Term Formula, but also being adept at basic mathematical operations such as exponentiation and multiplication. For example, in finding the 4th term in a sequence where the first term is 16 and the common ratio is \(-\frac{1}{3}\), we'd apply the formula directly: \[ a_4 = 16 \cdot \left(-\frac{1}{3}\right)^{3} \] Breaking it down, you first handle the exponentiation of the common ratio: \[ \left(-\frac{1}{3}\right) \times \left(-\frac{1}{3}\right) \times \left(-\frac{1}{3}\right) = -\frac{1}{27} \] Next, multiply this result by the first term to get: \[ a_4 = 16 \times -\frac{1}{27} = -\frac{16}{27} \]

• It's important to carefully follow the order of operations: exponentiation comes before multiplication.
• Perform each calculation step-by-step to avoid mistakes, particularly with negative numbers.
• Careful handling of fractions during both exponentiation and multiplication is crucial for accurate results.

By systematically applying these calculations, you can accurately identify specific terms in any geometric sequence.