Problem 30 - Algebra 2 and Trigonometry

Question

The first term of a geometric sequence is 1 and the 4 th term is \(27 .\) What is the 8 th term?

Step-by-Step Solution

Answer

The 8th term is 2187.

Step 1: Recall the formula for the nth term of a geometric sequence

The nth term of a geometric sequence can be expressed by the formula \(a_n = a_1 imes r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 2: Identify given values

From the problem, we know the first term \(a_1 = 1\) and the 4th term \(a_4 = 27\). We need to find the common ratio \(r\) using these values.

Step 3: Use the formula to find the common ratio

We use the formula for the 4th term: \(a_4 = a_1 imes r^{(4-1)} = r^3\). Thus, \(27 = 1 \times r^3\). Solve for \(r\) by taking the cube root: \(r = \sqrt[3]{27} = 3\).

Step 4: Calculate the 8th term using the common ratio

Now we use \(a_8 = a_1 imes r^{(8-1)} = 1 \times 3^{7}\). Calculate \(3^7\): \(3^7 = 2187\). Thus, \(a_8 = 2187\).

Key Concepts

nth term formula

In a geometric sequence, the position of each term in the sequence is defined by what we call the "nth term formula." This formula helps us determine the value of any term in a sequence, as long as we know the values of the first term and the common ratio. The formula is expressed as \( a_n = a_1 \times r^{(n-1)} \), where:

• \( a_n \) is the value of the term at position \( n \).
• \( a_1 \) is the first term in the sequence.
• \( r \) is the common ratio, the constant factor between consecutive terms.

In our exercise, the first term \( a_1 \) is 1. Using the 4th term (\( a_4 = 27 \)), we use the formula to solve for the common ratio \( r \). This process illustrates the fundamental utility of the nth term formula in unraveling the characteristics of a geometric sequence.

common ratio

The common ratio in a geometric sequence is a powerful component that defines how the terms in the sequence progress. It represents the factor by which each term will increase or decrease as you move forward or backward in the sequence. In math, we denote this constant factor by \( r \).

• If \( r > 1 \), the sequence increases.
• If \( 0 < r < 1 \), the sequence decreases.
• If \( r = 1 \), all terms are the same.
• If \( r < 0 \), the sequence alternates in sign.

In the provided exercise, we determined \( r = 3 \) by using the known 4th term value. Since \( r > 1 \), we can expect each subsequent term to be larger than the one before it. Understanding how to find and use the common ratio is essential for solving problems involving geometric sequences.

geometric series

A geometric series is the sum of the terms of a geometric sequence. Understanding the concept of a geometric series becomes essential when you want to find not just an individual term but the sum of multiple terms in the sequence. To compute the sum of the first \( n \) terms of a geometric series, we use the formula for the sum \( S_n \): \[ S_n = a_1 \frac{1-r^n}{1-r} \]provided \( r eq 1 \).

• \( S_n \) is the sum of the first \( n \) terms.
• \( a_1 \) is the first term.
• \( r \) is the common ratio.

In the exercise example, while it asks for the 8th term rather than a series sum, understanding the geometric series concept helps us appreciate how these terms accumulate and form part of a broader mathematical structure. This context is essential for solving more complex problems where the sum of terms is required.