Problem 38 - Precalculus Mathematics for Calculus

Question

The first term of a geometric sequence is \(3,\) and the third term is \(\frac{4}{3}\). Find the fifth term.

Step-by-Step Solution

Answer

The fifth term is \( \frac{16}{27} \).

Step 1: Understand the Problem

A geometric sequence has a constant ratio between consecutive terms. We know the first term, denoted as \( a_1 \), is 3, and the third term, \( a_3 \), is \( \frac{4}{3} \). We need to find the fifth term, \( a_5 \).

Step 2: Set Up the Formula for Terms

In a geometric sequence, terms are calculated using the formula \( a_n = a_1 \cdot r^{n-1} \), where \( r \) is the common ratio. We have\[a_1 = 3, \quad a_3 = \frac{4}{3}\]which gives us:\[a_3 = a_1 \cdot r^2 = 3 \cdot r^2 = \frac{4}{3}\]

Step 3: Solve for the Common Ratio

Rearrange the equation from Step 2 to solve for the common ratio, \( r \):\[3 \cdot r^2 = \frac{4}{3} \r^2 = \frac{4}{9} \r = \pm \frac{2}{3}\]Since both positive and negative ratios will yield valid sequences, we should check both when finding the fifth term.

Step 4: Calculate the Fifth Term with Positive Ratio

Using \( r = \frac{2}{3} \) and the formula:\[a_5 = a_1 \cdot r^4 = 3 \cdot \left(\frac{2}{3}\right)^4\]Simplify:\[a_5 = 3 \cdot \frac{16}{81} = \frac{48}{81} = \frac{16}{27}\]

Step 5: Calculate the Fifth Term with Negative Ratio

Using \( r = -\frac{2}{3} \) and the formula:\[a_5 = a_1 \cdot (-r)^4 = 3 \cdot \left(\frac{2}{3}\right)^4\]Calculate:\[a_5 = 3 \cdot \frac{16}{81} = \frac{48}{81} = \frac{16}{27}\]The fourth power of the negative ratio results in the same value as the positive ratio since it is an even power.

Step 6: Conclude with the Fifth Term

Since both calculations lead to the same result, regardless of the sign of \( r \), the fifth term \( a_5 \) is found to be \( \frac{16}{27} \).

Key Concepts

Common Ratio

In a geometric sequence, understanding the concept of the common ratio is essential. Every term after the first one is obtained by multiplying the previous term by this constant factor. This ratio remains the same throughout the sequence.

To find the common ratio, say you have the first term, denoted as \( a_1 \), equal to 3 and another term in the sequence, such as the third term \( a_3 \) equal to \( \frac{4}{3} \).

Use the formula:

• \( a_n = a_1 \cdot r^{n-1} \)

For example, to find \( r \) from the given values:

• \( a_3 = a_1 \cdot r^2 \)
• \( \frac{4}{3} = 3 \cdot r^2 \)

From here, solving gives \( r^2 = \frac{4}{9} \), so \( r = \pm \frac{2}{3} \).

This calculation shows that the common ratio can be either positive or negative, and that both forms are valid for geometric sequences.

Fifth Term

Finding the fifth term in a geometric sequence involves using the discovered common ratio with the sequence formula. Since two values for the common ratio were found, \( r = \frac{2}{3} \) and \( r = -\frac{2}{3} \), we will explore both.

The formula \( a_n = a_1 \cdot r^{n-1} \) helps calculate any term you desire. For the fifth term:

• Use \( a_5 = a_1 \cdot r^4 \)

Using the positive ratio \( \left( r = \frac{2}{3} \right) \):

• \( a_5 = 3 \cdot \left(\frac{2}{3}\right)^4 \)
• \( a_5 = 3 \cdot \frac{16}{81} \)
• \( a_5 = \frac{48}{81} = \frac{16}{27} \)

Interestingly, the same calculation applies even when using the negative ratio, \( r = -\frac{2}{3} \), due to the exponent being an even power. This property of powers makes the fifth term the same regardless of the sign of the ratio, leading to \( \frac{16}{27} \).

Sequence Formula

The sequence formula is a cornerstone for understanding geometric sequences. This formula, \( a_n = a_1 \cdot r^{n-1} \), is used to find any term \( n \) in a sequence. Each part of this formula serves a unique purpose:

• \( a_n \) represents the term you want to find.
• \( a_1 \) is the first term of the sequence, a starting point for calculations.
• \( r \) is the common ratio, which stays constant through the sequence.
• \( n-1 \) is the exponent applied to \( r \) to account for the term's position in the sequence.

For instance, if you need the third term in a sequence starting with 3 and a common ratio of \( \frac{2}{3} \), you'd substitute into the formula to find it:

\( a_3 = 3 \cdot \left(\frac{2}{3}\right)^2 \), which would help derive subsequent terms using the same framework, ensuring consistent and accurate results. Understanding and applying this formula allows for swift and efficient access to any term within a geometric sequence.