Problem 22 - 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. $$ 3,9,27,81, \dots $$

Step-by-Step Solution

Answer

The formula for the nth term is \(a_n = 3 \times 3^{(n-1)}\), and the ninth term is 19683.

Step 1: Identify the Pattern

Examine the given sequence: 3, 9, 27, 81, ... We notice that each term is obtained by multiplying the previous term by 3. This indicates that the sequence is geometric with a common ratio of 3.

Step 2: Write the General Term Formula

For a geometric sequence, the general term can be expressed as:\[ a_n = a_1 imes r^{(n-1)} \]where \(a_1\) is the first term and \(r\) is the common ratio.Here, \(a_1 = 3\) and \(r = 3\). Thus, the expression for the nth term is:\[ a_n = 3 imes 3^{(n-1)} \]

Step 3: Calculate the Ninth Term

Using the formula for the nth term:\[ a_n = 3 imes 3^{(n-1)} \]Substitute \(n = 9\) into the formula:\[ a_9 = 3 imes 3^{(9-1)} = 3 imes 3^8 \]Calculate \(3^8\):\[ 3^8 = 6561 \]Then, calculate \(a_9\):\[ a_9 = 3 imes 6561 = 19683 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are a kind of shorthand we use in mathematics to represent specific patterns, sequences, or operations. They consist of numbers, variables, and arithmetic operators. In the context of a geometric sequence, algebraic expressions serve as a formula to find terms without listing all preceding ones.
For example, in our sequence, the algebraic expression represents each term in the geometric sequence of 3, 9, 27, 81, and so forth. Here, the formula becomes more crucial as the sequence progresses because it allows us to determine any term in the sequence directly.

• The general form of the algebraic expression for a geometric sequence is: \[ a_n = a_1 \times r^{(n-1)} \]


• Here, \(a_1\) is the first term of the sequence, \(r\) is the common ratio, and \(n\) is the term number.

This expression simplifies the process of finding any term in the sequence, like the ninth term, without the need to multiply each term by the common ratio repeatedly.

Nth Term of a Sequence

The nth term of a sequence is a formula that allows you to find the term that is in the nth position of the sequence. For a geometric sequence like the one given, the nth term can be derived using its first term and the common ratio.

• The general formula is \( a_n = a_1 \times r^{(n-1)} \) which uses \(n\) to determine the place of the term you're trying to find.


• The nth-term formula eliminates the repetitive calculations by giving a direct way to calculate any desired term.

In our provided sequence, \(a_1 = 3\) and \(r = 3\). By plugging these values into the nth term formula, you can compute any specific term by simply changing \(n\). For instance, the ninth term is calculated by setting \(n\) to 9, which gives:

• Calculate \(3^{8} = 6561\).


• Then, compute \(a_9 = 3 \times 6561 = 19683\).

Knowing how to determine the nth term is especially advantageous when working with infinite sequences or those with very large indices.

Common Ratio

The common ratio is a fundamental concept in understanding geometric sequences. It is the factor by which we multiply one term of the sequence to arrive at the next term. Having a constant common ratio is what distinguishes a sequence as geometric.

• In our example, the common ratio, \(r\), is 3 because each term is three times the previous one.


• Knowing the common ratio allows you to construct the nth term formula: \( a_n = a_1 \times r^{(n-1)} \).

The beauty of understanding the common ratio is that it provides a way to predict the entire sequence as long as you know the first term. It simplifies the process and prevents errors that could occur from repeatedly calculating each term manually.

• Hence, once the first term and the common ratio are known, the entire sequence's behavior and progression can be predicted.


• This makes calculating terms like the ninth one simple and error-free.

Recognizing the common ratio allows for identifying the type of sequence—geometric, in this case—and simplifies working with sequences mathematically.