Problem 42 - Precalculus: Mathematics for Calculus

Question

Find the indicated term(s) of the geometric sequence with the given description. The fourth term is 12 and the seventh term is \(\frac{32}{9} .\) Find the first and \(n\) th terms.

Step-by-Step Solution

Answer

The first term is 40.5. The \( n \)th term is \( a_n = 40.5 \cdot \left(\frac{2}{3}\right)^{n-1} \).

Step 1: Identify the General Formula for a Geometric Sequence

The general formula for a geometric sequence is given by \( a_n = a \cdot r^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio. We will use this formula to find the variables \( a \) and \( r \).

Step 2: Write Equations for the Known Terms

We know the fourth term \( a_4 = 12 \), and using the sequence formula, this is \( a \cdot r^3 = 12 \). Similarly, the seventh term \( a_7 = \frac{32}{9} \) gives us \( a \cdot r^6 = \frac{32}{9} \).

Step 3: Solve the System of Equations

From the equations \( a \cdot r^3 = 12 \) and \( a \cdot r^6 = \frac{32}{9} \), divide the second equation by the first to eliminate \( a \). This yields \( r^3 = \frac{32}{9} \div 12 = \frac{32}{108} = \frac{8}{27} \). Taking the cube root, we find \( r = \frac{2}{3} \).

Step 4: Solve for the First Term \( a \)

Substitute \( r = \frac{2}{3} \) back into \( a \cdot r^3 = 12 \) to find \( a \). Calculating, \( a \cdot \left(\frac{8}{27}\right) = 12 \) gives us \( a = 12 \times \frac{27}{8} = 40.5 \).

Step 5: Generalize to Find the \( n \)th Term

Using the formula \( a_n = a \cdot r^{n-1} \), substitute \( a = 40.5 \) and \( r = \frac{2}{3} \). The \( n \)th term is \( a_n = 40.5 \cdot \left(\frac{2}{3}\right)^{n-1} \).

Key Concepts

First Term

In any geometric sequence, the first term, often denoted by \( a \), is crucial. It kickstarts the entire sequence, and every subsequent term in the sequence builds from it. In our exercise, the goal was to determine this first term from given terms further along in the sequence.
To find the first term, we used known values from the sequence. We had the fourth term as 12 and the seventh term as \( \frac{32}{9} \). By setting up equations with these terms in the general geometric formula, we reverse-calculated the first term.

• The fourth term provided the equation \( a \cdot r^3 = 12 \).
• The seventh term provided the equation \( a \cdot r^6 = \frac{32}{9} \).

Solving these equations helped us find the common ratio, and subsequent substitution helped us determine that the first term \( a \) is 40.5.

Common Ratio

The common ratio, denoted by \( r \), is a multiplier that allows you to move from one term to the next in a geometric sequence. It's a constant that remains the same throughout the sequence, and knowing it is crucial for understanding how the sequence grows or shrinks.
In the problem, we determined the common ratio by using the two known terms, the fourth and seventh terms. We extracted it by dividing the equations of these terms:

• Using the fourth term: \( a \cdot r^3 = 12 \).
• Using the seventh term: \( a \cdot r^6 = \frac{32}{9} \).

Dividing these equations canceled out the \( a \) and solved for \( r^3 \). We found \( r^3 = \frac{8}{27} \) and thus \( r = \left(\frac{8}{27}\right)^{1/3} = \frac{2}{3} \). This common ratio \( r = \frac{2}{3} \) is then used to predict any other terms in the sequence.

Nth Term

The nth term in a geometric sequence is an expression describing any term based on its position \( n \). It uses the sequence's first term and the common ratio. The formula for the nth term is given by \( a_n = a \cdot r^{n-1} \), where \( a_n \) is the nth term, \( a \) is the first term, and \( r \) is the common ratio.
This formula enables us to find any term's value in the sequence without having to manually calculate every single term before it. For instance, once we found \( a = 40.5 \) and \( r = \frac{2}{3} \), we could write:

• \( a_n = 40.5 \cdot \left(\frac{2}{3}\right)^{n-1} \)

By substituting any specific \( n \) into this formula, we can directly find the term value at that position. This approach significantly simplifies finding terms at arbitrary positions in the sequence.

Sequence Formula

The sequence formula for a geometric sequence defines the structure of the sequence. It is instrumental in both analyzing existing sequences and constructing new ones. The primary formula we use is \( a_n = a \cdot r^{n-1} \).
This formula encapsulates all crucial components:

• \( a \): the initial term that sets the sequence in motion.
• \( r \): the unchanging multiplier that defines movement through terms.
• \( n-1 \): representing the position within the sequence, showing how far along we are.

The formula is versatile and can solve problems related to finding specific terms (as we did in our exercise) or reconstructing sequences from partial information. By understanding each component, students can better visualize how a geometric sequence evolves. This sequence formula acts as a map—a guide to navigate through terms with precision and ease.