Problem 66 - Algebra 2

Question

The first term of a geometric sequence is 1 and its common ratio is \(6 .\) What is the sixth term? $$ \begin{array}{lllll}{\text { F. } 31} & {\text { G. } 3176} & {\text { H. } 7776} & {\text { J. } 46,656}\end{array} $$

Step-by-Step Solution

Answer

The sixth term of the geometric sequence is 7776. (Choice H)

Step 1: Understand the geometric sequence formula

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

Step 2: Plug in the known values into the formula

In the exercise, it's given that the first term of the sequence (\(a_1\)) is 1 and the common ratio (\(r\)) is 6. We want to find the 6th term (\(n = 6\)). Substituting these values into the equation gives us \(a_6 = 1 * 6^{(6-1)}\).

Step 3: Simplify to find the 6th term

To get the answer, compute \(6^{(6-1)}\) which simplifies to \(6^{5} = 7776\).

Key Concepts

Understanding the Common Ratio

A geometric sequence is a special type of sequence where each term after the first is found by multiplying the previous term by a constant number. This constant number is known as the **common ratio**.
In this particular exercise, the common ratio is given as 6, meaning each term is 6 times the previous term. Determining the common ratio is crucial because it dictates how quickly or slowly the terms in the sequence grow. A larger common ratio means the terms will grow significantly larger, whereas a smaller common ratio will lead to slower growth.
Understanding the role of the common ratio can help in predicting and calculating future terms in the sequence easily.

Utilizing the nth Term Formula

The nth term formula for a geometric sequence helps find any term in the sequence without the need to list all previous terms. The formula is given by \[ a_n = a_1 \times r^{n-1} \]where:

• \(a_n\) is the nth term you're looking for.
• \(a_1\) is the first term of the sequence.
• \(r\) represents the common ratio.
• \(n\) is the position of the term in the sequence.

This formula is a powerful tool because it allows you to quickly calculate any term in the sequence by plugging in the known values for the first term, the common ratio, and the number of terms.

Calculating a Specific Sequence Term

To calculate a specific term in a geometric sequence, follow these straightforward steps. Begin by identifying the known values: the first term \(a_1\), the common ratio \(r\), and the term position \(n\).
For instance, if you need to find the sixth term in a sequence where the first term \(a_1 = 1\) and the common ratio \(r = 6\), use the formula for the nth term:\[ a_6 = 1 \times 6^{6-1} \]**Step-by-step calculation:**

• Replace variables with known values: \( a_6 = 1 \times 6^{5} \).
• Calculate the power: \(6^5 = 7776\).
• Multiply by the first term: \(a_6 = 1 \times 7776 = 7776\).

Thus, the sixth term in this geometric sequence is 7776, option H in the exercise solution.