Problem 23 - Algebra 2 and Trigonometry

Question

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 12,6,3,1.5, \dots $$

Step-by-Step Solution

Answer

The ninth term is \(\frac{3}{64}\).

Step 1: Identify the Type of Sequence

Examine the given sequence: 12, 6, 3, 1.5, ... Notice that each consecutive term is half of the previous term. This indicates that the sequence is a geometric sequence where each term is multiplied by a common ratio.

Step 2: Determine the Common Ratio

To find the common ratio \(r\), divide any term by the previous term. For instance, \(r = \frac{6}{12} = \frac{1}{2}\). The common ratio \(r\) for this sequence is \(\frac{1}{2}\).

Step 3: Establish the General Formula for the Sequence

A geometric sequence can be represented by the formula \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio. Here, \(a_1 = 12\) and \(r = \frac{1}{2}\). Thus, the formula for the nth term is \(a_n = 12 \cdot \left(\frac{1}{2}\right)^{(n-1)}\).

Step 4: Calculate the Ninth Term

Substitute \(n = 9\) into the expression for \(a_n\). Thus, \(a_9 = 12 \cdot \left(\frac{1}{2}\right)^{(9-1)} = 12 \cdot \left(\frac{1}{2}\right)^8\). Calculate the result: \(\left(\frac{1}{2}\right)^8 = \frac{1}{256}\). Therefore, \(a_9 = 12 \cdot \frac{1}{256} = \frac{12}{256} = \frac{3}{64}\).

Key Concepts

Common Ratio

In a geometric sequence, understanding the **common ratio** is pivotal. This ratio is the factor by which we multiply each term to get the next term in the sequence. In simpler words, the common ratio acts like a rule that helps us determine what the next number will be.
To find the common ratio, you can divide any term in the sequence by its previous term. Let's look at the sequence: 12, 6, 3, 1.5, ...

• Take the second term and divide it by the first: \( r = \frac{6}{12} = \frac{1}{2} \).
• Repeat with the next pair: \( \frac{3}{6} = \frac{1}{2} \), and similarly for \( \frac{1.5}{3} = \frac{1}{2} \).

Each division yields the same result \( \frac{1}{2} \), confirming that \( \frac{1}{2} \) is the common ratio of this sequence.
Next time you see a sequence, try discovering its common ratio by this method.

Nth Term Formula

Once the common ratio is identified, the **nth term formula** becomes your best friend in finding any specific term in a geometric sequence. This formula allows you to calculate the value of the term found at any position "n" without listing all the sequence members.
The general formula for the nth term of a geometric sequence is:\[ a_n = a_1 \cdot r^{(n-1)} \]Here:

• \( a_n \) is the nth term.
• \( a_1 \) represents the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term you wish to find.

In our given sequence, \( a_1 = 12 \) and \( r = \frac{1}{2} \), thus the formula becomes:\[ a_n = 12 \cdot \left(\frac{1}{2}\right)^{(n-1)} \]By using this formula, you can quickly find any term like the fifth or the twentieth, without recalculating all previous terms.

Sequence Identification

Identifying a sequence's type is crucial when approaching any series of numbers. This allows us to decide on using the right formulas and methods to gain insights into the sequence properties.
The sequence 12, 6, 3, 1.5, ... can be identified as a **geometric sequence**. This type of sequence is characterized by a constant ratio between successive terms. In comparison, an arithmetic sequence has a constant difference between terms.
To identify if a sequence is geometric:

• Calculate the ratio of consecutive terms. If this ratio is constant and non-zero, your sequence is geometric.
• Look for patterns. If terms are repeatedly halving, doubling, or tripling, you're likely dealing with a geometric sequence.

Spotting and categorizing a sequence assists in solving problems and understanding mathematical patterns and ideas effectively.