Question

In Exercises \(7-10,\) find the \(6^{\text {th }}\) and \(n^{\text {th }}\) terms of the geometric sequence. $$ 1,3,9, \ldots $$

Short answer

Question: Find the 6th term and the general formula for the nth term of the geometric sequence 1, 3, 9, 27, ... Answer: The 6th term of the sequence is 243, and the general formula for the nth term is \(a_n = 1 \cdot 3^{n-1}\).

Step-by-step solution

Identify the common ratio

To find the common ratio (r), divide any consecutive terms in the sequence. In this case: $$ r = \frac{3}{1} = 3 $$ The common ratio is 3.

Find the 6th term of the sequence

To find the 6th term, we use the geometric sequence formula: $$ a_n = a_1 \cdot r^{n-1} $$ Where \(a_n\) is the nth term, \(a_1\) is the first term, and n is the term number. In our case, \(a_1 = 1\), \(r = 3\), and \(n = 6\): $$ a_6 = 1 \cdot 3^{6-1} = 1 \cdot 3^5 = 1 \cdot 243 = 243 $$ The 6th term is 243.

Find the general formula for the nth term of the sequence

Now that we know the common ratio and the first term, we can write the general formula for the nth term: $$ a_n = 1 \cdot 3^{n-1} $$ This formula allows us to find any term in the geometric sequence.

Key concepts

Understanding the Common Ratio

The common ratio is a fundamental element of any geometric sequence. It is the factor by which each term is multiplied to get the next term. In simpler terms, it shows how much the sequence "grows" from one term to the next. For the given sequence, 1, 3, 9, ..., you determine the common ratio by dividing any term by the preceding term.

• In this sequence, when you divide 3 by 1, you get 3.
• Similarly, dividing 9 by 3 results in 3.

Thus, the common ratio is 3, indicating that each term is three times the previous term. Understanding the common ratio is crucial for exploring and predicting the behavior of geometric sequences.

Calculating the nth Term

The "nth term" refers to finding a specific term in a sequence without listing all the preceding terms. To calculate the nth term of a geometric sequence, use the formula:\[a_n = a_1 \cdot r^{n-1}.\]Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence. For the sequence 1, 3, 9, ..., suppose you want to find the 6th term:

• Start with \(a_1 = 1\).
• Use the common ratio, \(r = 3\).
• Set \(n = 6\).

Substitute these values into the formula: \(a_6 = 1 \cdot 3^{6-1} = 243\). This method allows you to efficiently find any term in the sequence without having to calculate all previous terms.

Geometric Sequence Formula

The geometric sequence formula is a powerful tool used for generating any term in a sequence based on the sequence’s starting point and growth pattern. The formula \( a_n = a_1 \cdot r^{n-1} \) incorporates:

• \(a_1\), the first term.
• \(r\), the common ratio.
• \(n\), the term number you wish to find.

This formula reveals the relationship between terms and helps predict future terms by scaling the initial term \(a_1\) by the repeated factor \(r\) raised to a power. For instance, knowing \(a_1 = 1\) and \(r = 3\) in our example means any term can be determined using \[ a_n = 1 \cdot 3^{n-1} \]. This approach is efficient for solving sequence problems and predicting sequences' long-term behavior, especially when terms grow or shrink dramatically due to a large or small common ratio.