Problem 27 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{7} \text { for } \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \dots $$

Step-by-Step Solution

Answer

The seventh term of the sequence is 2.

Step 1: Identify the First Term and Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. The first term of this sequence is \( a_1 = \frac{1}{32} \). To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{1/16}{1/32} = 2 \).

Step 2: Use the Formula for the nth Term

The formula for the nth term of a geometric sequence is \( a_n = a_1 \times r^{(n-1)} \). Here, \( a_1 = \frac{1}{32} \), \( r = 2 \), and \( n = 7 \). Substitute these values into the formula to find \( a_7 \).

Step 3: Calculate the Seventh Term

Substitute the known values into the formula: \( a_7 = \frac{1}{32} \times 2^{(7-1)} \). This simplifies to \( a_7 = \frac{1}{32} \times 2^6 \).

Step 4: Simplify the Expression

Calculate \( 2^6 \), which is \( 64 \). Then, multiply: \( a_7 = \frac{1}{32} \times 64 \). This simplifies to \( a_7 = 2 \).

Key Concepts

Common Ratio

In a geometric sequence, the difference between consecutive terms isn't based on addition or subtraction but rather multiplication by a fixed number known as the common ratio. This unique feature sets geometric sequences apart from arithmetic sequences, where the difference is constant by addition. For example, in the sequence \( \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \dots \), you calculate the common ratio \( r \) by dividing any term by its preceding term.

To find \( r\), simply divide the second term by the first term:

• First term \( a_1 = \frac{1}{32} \)
• Second term \( a_2 = \frac{1}{16} \)

Using these values, the common ratio is computed as \( r = \frac{1/16}{1/32} = 2 \). This means that each term in the sequence is twice its preceding term. The common ratio is critical because it helps determine the behavior of the sequence and is used extensively in calculations involving geometric sequences.

Nth Term Formula

To find any term in a geometric sequence, we use the nth term formula. This formula provides a straightforward method to calculate any term based on the positions of the terms and the sequence’s characteristics. The general formula for the nth term is: \[ a_n = a_1 \times r^{(n-1)} \]

Here’s what each symbol stands for:

• \( a_n \): the term we want to find
• \( a_1 \): the first term in the sequence
• \( r \): the common ratio
• \( n \): the term's position in the sequence

Using this formula allows us to determine any term without calculating each preceding term. For example, to find the 7th term in our sequence, we would set \( a_1 = \frac{1}{32} \), \( r = 2 \), and \( n = 7 \).
Substituting into the formula, we calculate \( a_7 = \frac{1}{32} \times 2^{(7-1)} \), which simplifies further. This formula is indispensable for efficiently working with geometric sequences.

First Term in Geometric Sequence

The starting point in any geometric sequence is crucial as it acts as the foundation for all subsequent terms. Denoted as \( a_1 \), the first term determines the baseline value from which the sequence commences. In the example sequence \( \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \dots \), the first term is \( a_1 = \frac{1}{32} \).

The first term plays an essential role in the process of determining other terms as it is multiplied by the common ratio raised to a power, according to its position. For example, in the nth term formula \( a_n = a_1 \times r^{(n-1)} \), the term \( a_1 \) determines the constant value related to all terms of the sequence.
Always identifying the first term in a geometric sequence is the stepping stone for successfully leveraging the common ratio and nth term formula in problem-solving.