Problem 14 - Algebra for College Students

Question

The 7 th term of \(2,6,18,54, \ldots\) 1458

Step-by-Step Solution

Answer

The 7th term of the sequence is 1458.

Step 1: Understand the Sequence

The given sequence is 2, 6, 18, 54, ... , which suggests a geometric progression. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.

Step 2: Identify the Common Ratio

To find the common ratio (r), divide the second term by the first term: \( r = \frac{6}{2} = 3 \). Verify this ratio by dividing the third term by the second term: \( \frac{18}{6} = 3 \) and similarly, \( \frac{54}{18} = 3 \). Thus, the common ratio is confirmed as 3.

Step 3: Use the Geometric Sequence Formula

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

Step 4: Substitute Values to Find the 7th Term

Given \( a_1 = 2 \), \( r = 3 \), and \( n = 7 \), substitute these values into the formula: \( a_7 = 2 \cdot 3^{7-1} = 2 \cdot 3^6 \).

Step 5: Calculate the Power of the Ratio

Calculate \( 3^6 \): \( 3^6 = 729 \).

Step 6: Multiply to Find the Seventh Term

Now, compute \( a_7 = 2 \cdot 729 = 1458 \). Thus, the 7th term of the sequence is 1458.

Key Concepts

common ratio

In geometric sequences, a constant value known as the "common ratio" determines how each term relates to the previous one. We find the common ratio by dividing any term in the sequence by the term before it. For example, consider the sequence 2, 6, 18, 54, ... . You can figure out the common ratio, denoted as \( r \), by dividing the second term by the first one: \( r = \frac{6}{2} = 3 \).

• Confirm this by dividing the third term, 18, by the second term, 6: \( \frac{18}{6} = 3 \).
• Do the same with 54 and 18 to further verify: \( \frac{54}{18} = 3 \).

The value of 3 confirms the sequence has a common ratio of 3. Knowing the common ratio is crucial because it allows you to continue a sequence and find any other term.

nth term formula

The nth term formula is a vital tool in geometric sequences, enabling you to find any term in the sequence without listing all the preceding terms.
The formula for finding the nth term \( a_n \) is \( a_n = a_1 \cdot r^{n-1} \):

• \( a_1 \) is the first term in the sequence.
• \( r \) is the common ratio.
• \( n \) represents the position of the term in the sequence you want to find.

Let's see this in action. For our sequence \( 2, 6, 18, 54, \ldots \), if you want to find the 7th term, plug in the values: \( a_7 = 2 \cdot 3^{7-1} \). This calculation involves noting that the first term \( a_1 = 2 \), the common ratio \( r = 3 \), and the term number \( n = 7 \). Running the math gives \( a_7 = 2 \cdot 729 = 1458 \), showing how efficiently the formula lets you pinpoint any position.

geometric progression

A geometric progression is a type of sequence where each term is derived from the previous one by multiplying by the same non-zero constant, the common ratio.
Understanding geometric progression helps in realizing how sequences unfold, making it easier to predict and calculate any term.

• It's characterized by its regularity, much like our sequence \( 2, 6, 18, 54, \ldots \).
• In any geometric sequence, given the first term and the common ratio, you can generate subsequent terms.
• This type of sequence often appears in growth patterns, like population growth, finance (compound interest), and models of exponential growth.

For example, starting with 2 and a common ratio of 3, multiply successively to get 6, 18, and so on. These predictable steps showcase how versatile and practical geometric progressions are in various contexts, helping to easily manage and analyze numeric patterns.