Problem 24 - Beginning and Intermediate Algebra

Question

Find the general term of each geometric sequence. $$4,12,36,108, \dots$$

Step-by-Step Solution

Answer

The general term of the geometric sequence is \(a_n = 4 \cdot 3^{n-1}\).

Step 1: Identify the first term and the common ratio

The first term in the sequence is \(4\). To find the common ratio, we can divide the second term by the first term or any subsequent term by the previous term. In this case, we divide the second term (\(12\)) by the first term (\(4\)): \[r = \frac{12}{4} = 3\] The common ratio is \(3\).

Step 2: Use the formula to find the general term

Now that we have the first term (\(a_1 = 4\)) and the common ratio (\(r = 3\)), we can find the general term, \(a_n\), using the formula: \[a_n = a_1 \cdot r^{n-1}\] Substitute the values of \(a_1\) and \(r\) into the formula: \[a_n = 4 \cdot 3^{n-1}\] This is the general term of the given geometric sequence.

Key Concepts

general term formula

In a geometric sequence, the general term formula is your essential tool to find any term in the sequence without listing all the previous terms. This formula is expressed as \(a_n = a_1 \cdot r^{n-1}\). Here, \(a_n\) represents the \(n\)-th term of the sequence. \(a_1\) is the first term, and \(r\) is the common ratio. By applying this formula, you can easily navigate through a sequence, knowing exactly which term you are looking for. Let's break it down further:- **\(a_1\)**: This is the first term of your sequence. In the exercise example, \(a_1 = 4\).- **\(r^{n-1}\)**: Here, \(r\) is the common ratio, and \(n-1\) is the exponent showing how many times you multiply \(a_1\) by \(r\) to find the \(n\)-th term. For our exercise, the formula \(a_n = 4 \cdot 3^{n-1}\) helps find any term in the sequence without repeatedly multiplying.

common ratio

Understanding the common ratio in a geometric sequence is crucial because it defines the consistent pattern of how the sequence progresses. In our exercise, to find the common ratio \(r\), you divide any term by its preceding term. This ratio remains constant throughout the sequence.In the given sequence \(4, 12, 36, 108, \ldots\), we computed \(r\) as follows:- Divide the second term by the first term: \(\frac{12}{4} = 3\). Thus, \(r = 3\).The role of the common ratio includes:

• Determining the multiplicative factor from one term to the next.
• Being the base in the exponent part of the general term formula.
• Helping validate if a sequence is truly geometric.

A constant, unchanging ratio is what makes a sequence geometric, identified correctly in this step of the exercise.

sequence patterns

Recognizing sequence patterns involves determining what's changing from one term to the next. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number, called the common ratio. Visually and mathematically, this replacement or progression will always happen consistently: - Start with a given first term. - Multiply it by the common ratio to reach the next. - Apply this process repeatedly to generate more terms. For example, after identifying the consistent multiplication of term by the common ratio, you can predict the sequence: starting at 4, multiplying by 3 gives 12, then 36, and so on. Pattern recognition in such sequences allows not only for predicting future terms but also for unequivocal identification of any term's exact position or value in the sequence using its general formula. Understanding these patterns aids in solving more complex problems where sequences might play a part.