Problem 6 - Algebra and Trigonometry

Question

Find the \(n\)th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$ a=-6, \quad r=3 $$

Step-by-Step Solution

Answer

The fourth term is -162.

Step 1: Understand the formula for the nth term

For any geometric sequence, the formula for the nth term is given by \( a_n = a \cdot r^{(n-1)} \), where \( a \) is the first term and \( r \) is the common ratio.

Step 2: Plug in the values for the nth term

We are asked to find the fourth term of the sequence, so we'll use \( n = 4 \). Plugging the values into the formula, we get \( a_4 = -6 \cdot 3^{(4-1)} \).

Step 3: Simplify the exponent

Simplify the exponent by calculating \( 4-1 = 3 \). Then the expression becomes \( a_4 = -6 \cdot 3^3 \).

Step 4: Calculate the common ratio raised to the power

Calculate \( 3^3 \), which is \( 27 \). So we can substitute this back into our expression: \( a_4 = -6 \cdot 27 \).

Step 5: Calculate the nth term

Perform the multiplication: \( -6 \times 27 = -162 \). Therefore, the fourth term \( a_4 \) in the geometric sequence is \(-162\).

Key Concepts

Understanding the nth Term Formula in Geometric Sequences

In geometric sequences, each term is derived from the previous one by multiplying with a constant known as the "common ratio". To efficiently find any term in the sequence, we use the nth term formula. The formula is:

\[ a_n = a \cdot r^{(n-1)}\]
Here, \(a_n\) is the nth term we want to calculate, \(a\) is the first term, and \(r\) is the common ratio. The expression \((n-1)\) is used as the exponent of \(r\) because the first term is already \(a\), and the multiplication starts from the second term. Understanding this formula allows us to jump directly to any position within the sequence without listing all preceding terms. It is both a time-saver and a tool for unlocking the inherent structure of geometric sequences.

The Importance of the Common Ratio

The common ratio is a crucial element in any geometric sequence. Denoted as \(r\), it represents the factor by which each term is multiplied to yield the next term. If you know the first term and the common ratio, you can determine the entire sequence.

Imagine a sequence starts at \(-6\) and has a common ratio of \(3\). This means that each following term is three times the previous one:

• First term \((a)\): \(-6\)
• Second term: \(-6 \cdot 3 = -18\)
• Third term: \(-18 \cdot 3 = -54\), and so on.

The sign and value of \(r\) are important:

• If \(r > 1\), the sequence grows.
• If \(0 < r < 1\), the sequence diminishes.
• If \(r = 0\), all terms become zero.
• A negative \(r\) means the terms alternate in sign.

Thus, knowing the common ratio gives insight into the sequence's behavior.

Steps in Sequence Calculation

Calculating terms in a geometric sequence involves several clear steps, which become straightforward once you understand the fundamentals:

• Identify Known Values: Start by finding the first term \(a\) and the common ratio \(r\).
• Apply the nth Term Formula: Use \( a_n = a \cdot r^{(n-1)} \) for the term you are interested in.
• Simplify the Exponent: Calculate \((n-1)\) and use the result as the exponent for \(r\).
• Evaluate the Power of the Common Ratio: Raise \(r\) to the power of \((n-1)\) to find how much larger or smaller it becomes.
• Compute the nth Term: Multiply the result from the previous step by the first term \(a\) to find \(a_n\).

These steps ensure you can find any term of the sequence with precision. Using the given exercise information, following these steps leads us to understand that the fourth term results in \(-162\). This process exemplifies consistent results, whether the sequence is increasing or changing signs.