Problem 28 - Algebra and Trigonometry

Question

Find the indicated term of each geometric sequence. 8th term of \(1,3,9, \ldots\)

Step-by-Step Solution

Answer

The 8th term is 2187.

Step 1: Identify the first term and the common ratio

Observe the given sequence: \(1, 3, 9, \ldots\). The first term \(a\) is 1. To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{3}{1} = 3\).

Step 2: Use the nth term formula for geometric sequences

The formula for the nth term of a geometric sequence is: \( a_n = a \cdot r^{n-1} \).

Step 3: Substitute the known values into the formula

To find the 8th term, substitute \( a = 1 \), \( r = 3 \), and \( n = 8 \) into the formula: \( a_8 = 1 \cdot 3^{8-1} \).

Step 4: Simplify the exponentiation

Calculate \( 3^{7} \): \( 3^7 = 2187 \).

Step 5: Final calculation

Multiply the first term by the result of the exponentiation: \( a_8 = 1 \cdot 2187 = 2187 \).

Key Concepts

Common Ratio

To fully understand geometric sequences, it's crucial to grasp the concept of the 'common ratio'. The common ratio, denoted as \(r\), is the factor by which each term in a geometric sequence is multiplied to obtain the next term. In the given sequence \(1, 3, 9, \ldots\), you can determine the common ratio by dividing the second term by the first term: \(r = \frac{3}{1} = 3\). This means that each term is three times the previous term. Recognizing the common ratio is the first step in solving most geometric sequence problems. It provides a consistent factor that drives the progression of the sequence, making it predictable.

Nth Term Formula

Once you identify the common ratio, the next important concept is the 'nth term formula'. This formula helps you find any specific term in a geometric sequence. The nth term \(a_n\) of a geometric sequence can be calculated using the formula: \( a_n = a \cdot r^{n-1} \). Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number you want to find. For example, to find the 8th term of the sequence \(1, 3, 9, \ldots\), you would set \(a = 1\), \(r = 3\), and \(n = 8\). Substituting these values into the formula, you get: \( a_8 = 1 \cdot 3^{8-1} \). Understanding this formula allows you to find any term in the sequence without having to list all the previous terms.

Exponentiation

The final step in solving geometric sequence problems usually involves 'exponentiation', which is the process of raising a number to a power. In our example, finding \( a_8 = 1 \cdot 3^{8-1} \) requires calculating \( 3^7 \). Exponentiation is denoted as \( b^e \), where \(b\) is the base and \(e\) is the exponent. For \(3^7\), the calculation is: \( 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 2187 \). Therefore, \( a_8 = 1 \cdot 2187 = 2187 \). Exponentiation simplifies the process of finding terms in geometric sequences, especially as the term position \(n\) increases. Mastering exponentiation lets you handle large values efficiently, which is particularly useful in geometric sequences.