Problem 20 - College Algebra

Question

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\)to find \(a_{7},\) the seventh term of the sequence. $$12,6,3, \frac{3}{2}, \ldots$$

Step-by-Step Solution

Answer

The seventh term of the sequence is \(a_{7}=12 \times (0.5)^{6} = 0.1875.\)

Step 1: Finding the First Term (\(a_{1}\)) and the Common Ratio (\(r\))

Firstly, identify the first term \(a_{1}\) and the common ratio \(r\) in the sequence. The first term \(a_{1}\) is given as 12, and the common ratio \(r\) is the factor by which consecutive terms increase or decrease. This can be found by dividing any term by the term before it. In this case, \(r=6/12=0.5\)

Step 2: Formulate the General Sequence Formula

Now that we know \(a_{1}\) and \(r\), we can write the general sequence formula, which is \(a_{n}=a_{1} \times r^{n-1}\). Substituting \(a_{1}=12\) and \(r=0.5\) into the equation, the formula for this specific geometric sequence becomes \(a_{n}=12 \times (0.5)^{n-1}\).

Step 3: Find the Seventh Term (\(a_{7}\))

To find the 7th term \(a_{7}\) of the sequence, substitute \(n=7\) into the equation \(a_{n}=12 \times (0.5)^{n-1}\) to get \(a_{7}=12 \times (0.5)^{6}\). Calculate this expression to find the value of \(a_{7}\).

Key Concepts

General Term Formula

Understanding the general term formula for a geometric sequence is a core component of grasping sequences and series in mathematics. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general term formula, denoted as \( a_n \), allows you to find any term in the sequence without listing all the preceding terms.

The general term formula for a geometric sequence is \( a_n = a_1 \times r^{(n-1)} \), where:

• \( a_n \) is the nth term you want to find.
• \( a_1 \) is the first term in the sequence.
• \( r \) is the common ratio between the terms.
• \( n \) is the term number.

This formula is a powerful tool because it shows that each term in the sequence is a product of the initial term and a power of the common ratio. To apply this to a particular sequence, simply insert the appropriate values for \( a_1 \), \( r \), and \( n \).

Common Ratio

The common ratio in a geometric sequence is the factor that consistently relates consecutive terms. It is denoted as \( r \) and can be found by dividing any term in the sequence by its preceding term (except for the first term since there is no term before it to divide by).

It's important to note that the common ratio can be positive or negative, which affects the pattern of the sequence, and it can also be a fraction, as seen in our example sequence. If the common ratio is greater than 1, the terms in the sequence grow larger; if the ratio is between 0 and 1, the terms get smaller; and if the ratio is negative, the terms alternate in sign.

In the sequence \(12, 6, 3, \frac{3}{2}, \ldots \), the common ratio is calculated as follows: \( r = \frac{6}{12} = 0.5 \). This implies that each term is half the term before it. Recognizing the common ratio allows you to predict the direction and pattern of the sequence over time without computing every individual term.

Sequence and Series

A sequence is an ordered list of numbers that follow a specific pattern, while a series is the sum of the terms of a sequence. In the context of geometric sequences, each term is determined by multiplying the previous term by the constant common ratio, and this pattern continues indefinitely.

It's critical to differentiate between these two concepts:

• A sequence refers to the individual elements that follow a rule (e.g., every term after the first is obtained by multiplying the previous term by the common ratio).
• A series represents the sum of these elements, which can be finite or infinite depending on whether the sequence has an end or not. For a geometric series, there's a handy formula for the sum, especially for an infinite series with a common ratio between -1 and 1 (where the series converges).

Geometric Series vs. Geometric Sequence

While the sequence \(12, 6, 3, \frac{3}{2}, \ldots \) lists the terms, the series would be the expression \(12 + 6 + 3 + \frac{3}{2} + \ldots \), which sums the terms. In the classroom or on homework, it's common to deal with both finding individual terms (using the general term formula) and summing parts of or entire sequences (dealing with series). Recognizing when to use the right formula is key to solving these problems correctly.