Problem 52 - Algebra for College Students

Question

$$ 2,-6,18,-54, \ldots $$ No sum

Step-by-Step Solution

Answer

The next term in the sequence is 162.

Step 1: Identify the Pattern

Observe the progression of the sequence: 2, -6, 18, -54, ...Notice that each term is obtained by multiplying the previous term by a specific number. Begin by dividing the second term by the first: \(-6 \div 2 = -3\). Similarly, check the next pair: 18 \div -6 = -3. And check the next one: \(-54 \div 18 = -3\).Each term is obtained by multiplying the previous term by \(-3\). This indicates that the sequence is a geometric sequence with a common ratio \(r = -3\).

Step 2: Use the Formula for the nth Term

In a geometric sequence, the nth term \(a_n\) can be found using the formula:\[ a_n = a_1 \, r^{(n-1)} \]where \(a_1\) is the first term, and \(r\) is the common ratio. In this sequence, \(a_1 = 2\) and \(r = -3\).

Step 3: Find the Next Term in the Sequence

Apply the common ratio multiplication to find the next term.The fourth term is \(-54\). To find the fifth term, multiply this term by \(-3\):\[ -54 \times (-3) = 162 \].So the next term in the sequence is 162.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key factor that helps us understand how each term relates to the previous one. The common ratio, often denoted as \( r \), is a constant factor that we multiply each term by to get the next term in the sequence.
In the sequence you're working with, "2, -6, 18, -54, ...," each term is generated by multiplying the preceding term by \(-3\).
This constant, \(-3\), is your common ratio.

• Find the common ratio by dividing any term by the one before it.
• Example: \(-6 \div 2 = -3\), \( 18 \div -6 = -3\).
• It must remain constant throughout the sequence.

Identifying the common ratio is crucial because it underpins the repeating pattern of the sequence.

Nth Term Formula

The nth term formula for a geometric sequence allows us to find any term without needing to calculate all the previous ones. The formula is:\[ a_n = a_1 \, r^{(n-1)} \]Here, \(a_n\) represents the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.
For our sequence, the first term \(a_1\) is 2, and the common ratio \(r\) is \(-3\).
Let's see how it works in practice:

• Plug the values into the formula: \(a_n = 2 \times (-3)^{(n-1)}\).
• To find the third term: \(a_3 = 2 \times (-3)^{2} = 2 \times 9 = 18\).
• Notice how repeating this process helps predict any term in the sequence.

This formula is especially helpful when you need to find a term further along without calculating each preceding term one by one.

Sequence Pattern

Understanding the pattern in a sequence is essential for recognizing the underlying mathematical relationship.
Geometric sequences are unique because they have a multiplicative pattern, created by a common ratio. Here's how to grasp the sequence pattern:

• Start by identifying if it's a geometric sequence using the common ratio method.
• Notice the repetition: multiply by \(r\) to advance from one term to the next.
• A geometric sequence either grows or shrinks exponentially due to this fixed ratio.

In the given sequence, "2, -6, 18, -54, ...," you recognize the consistent multiplicative pattern by applying \(-3\) to each term.
This discovery simplifies predicting subsequent terms and understanding the sequence's behavior.