Problem 18 - Beginning and Intermediate Algebra

Question

Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=-5, r=-\frac{1}{3} ; a_{4}$$

Step-by-Step Solution

Answer

The general term of the geometric sequence is \(a_m = (-5) \cdot \left(-\frac{1}{3}\right)^{m-1}\) and the 4th term is \(a_4 = \frac{5}{27}\).

Step 1: Write down the given information

We are given the following information for the geometric sequence: - The first term \(a_1 = -5\) - The common ratio \(r = -\frac{1}{3}\) Our goal is to find the general term \(a_m\) and the fourth term in the sequence, \(a_4\).

Step 2: Find the general term of the geometric sequence

Using the formula for the general term of a geometric sequence, we can find \(a_m\): $$a_m = a_1 \cdot r^{m-1}$$ Plugging in the given values for the first term and common ratio, we get: $$a_m = (-5) \cdot \left(-\frac{1}{3}\right)^{m-1}$$ So, the general term of the given geometric sequence is: $$a_m = (-5) \cdot \left(-\frac{1}{3}\right)^{m-1}$$

Step 3: Find the 4th term of the geometric sequence

Now that we have the general term, we can find the fourth term \(a_4\) by plugging in \(m=4\) into the formula: $$a_4 = (-5) \cdot \left(-\frac{1}{3}\right)^{4-1}$$ Simplify the exponent: $$a_4 = (-5) \cdot \left(-\frac{1}{3}\right)^3$$ Now, we calculate the power in the parentheses: $$a_4 = (-5) \cdot -\frac{1}{27}$$ Finally, multiply the two terms together: $$a_4 = \frac{5}{27}$$ So, the fourth term of the given geometric sequence is \(a_4 = \frac{5}{27}\).

Key Concepts

General Term

The general term is a crucial component of geometric sequences. It allows us to find any term in the sequence without having to list all the previous ones.
Understanding the formula for the general term is essential:

• The general term, also known as the nth term, is given by the formula \(a_m = a_1 \cdot r^{m-1}\)
• Here, \(a_1\) represents the first term, \(r\) is the common ratio, and \(m\) is the term number we are interested in.

Let's quickly dissect this formula: The first term \(a_1\) sets the sequence's starting point.
The factor \(r^{m-1}\) reflects the exponential growth or decay defined by the common ratio, applied \(m-1\) times as you progress through the sequence.

The beauty of the general term formula is its efficiency.
With it, you can directly find any term in the sequence without a lengthy process, once you have the necessary initial details.

Common Ratio

The common ratio in a geometric sequence significantly influences how the sequence progresses.
This ratio, often denoted as \(r\), is the constant factor between consecutive terms.
It shapes the overall nature of the sequence, which can either grow, decay, or fluctuate based on its value.

• If the common ratio \(r\) is greater than 1, the sequence experiences exponential growth.
• If \(r\) is between 0 and 1, the sequence decays and the terms get smaller.
• When \(r\) is negative, the sequence alternates signs, leading to an oscillating pattern.
• If \(r\) equals 1, each term is the same, producing a constant sequence.

In our example, the common ratio is \(-\frac{1}{3}\), which implies alternating the sign and diminishing the magnitude with every step forward.
This emphasizes the unique oscillating nature and gradual decay of this specific sequence.

Understanding the role of the common ratio is pivotal.
It helps predict and comprehend the behavior of the entire sequence, providing a clearer picture of its dynamics.

Sequence Term Calculation

Finding a specific sequence term in a geometric sequence requires using the general term formula as a tool.
By substituting the term's position number, you conveniently calculate its value.
Consider the following approach:

• First, identify your sequence's starting point and common ratio, which are essential components.
• Substitute these known values into the general term formula.
• Select the term number \(m\) you’re looking to find and plug it into the formula.

For instance, to find the fourth term \(a_4\) in our sequence, note how we applied this step-by-step:

• The formula \(a_4 = (-5) \cdot \left(-\frac{1}{3}\right)^{3}\) emerges.
• Negatively increasing the power, \(-\frac{1}{3}\) raised to the third power yields \(-\frac{1}{27}\).
• Multiplying by the initial term \(-5\), produces \(\frac{5}{27}\).

This methodical approach ensures accurate calculation.
Thus, effectively unlocking any desired term within the sequence by leveraging and understanding these mathematical foundations.