Problem 28 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 7, \frac{14}{3}, \frac{28}{9}, \frac{56}{27}, \dots $$

Step-by-Step Solution

Answer

The common ratio is \( \frac{2}{3} \), the fifth term is \( \frac{112}{81} \) and the nth term is \( 7 \cdot \left( \frac{2}{3} \right)^{n-1} \).

Step 1: Identify the Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This sequence starts with 7, then \( \frac{14}{3} \), then \( \frac{28}{9} \), and then \( \frac{56}{27} \).

Step 2: Calculate the Common Ratio

The common ratio \( r \) can be found by dividing the second term by the first term. So \( r = \frac{14/3}{7} = \frac{2}{3} \). Let's verify by checking \( \frac{28/9}{14/3} = \frac{2}{3} \) and \( \frac{56/27}{28/9} = \frac{2}{3} \). Thus, the common ratio \( r = \frac{2}{3} \).

Step 3: Calculate the Fifth Term

In a geometric sequence, the nth term is found by \( a_n = a_1 \cdot r^{n-1} \). We want the fifth term, so \( a_5 = 7 \cdot \left( \frac{2}{3} \right)^{4} = 7 \cdot \frac{16}{81} = \frac{112}{81} \).

Step 4: Derive the General Formula for the nth Term

The general formula for the nth term of a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \). Since \( a_1 = 7 \) and \( r = \frac{2}{3} \), we have \( a_n = 7 \cdot \left( \frac{2}{3} \right)^{n-1} \).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key factor. It is a constant value that you multiply by each term to find the next term in the sequence. To determine the common ratio in the sequence \(7, \frac{14}{3}, \frac{28}{9}, \frac{56}{27}, \ldots\), start with the first two terms:

• First term: \(7\)
• Second term: \(\frac{14}{3}\)

Find the common ratio \(r\) by dividing the second term by the first:\[r = \frac{\frac{14}{3}}{7} = \frac{2}{3}\]
This calculation shows you that every term is \(\frac{2}{3}\) times the previous term. As verified by also dividing the third term by the second, and the fourth by the third, the common ratio remains consistent. This consistent value \(\frac{2}{3}\) affects how the sequence progresses.

nth Term of a Sequence

The nth term of a geometric sequence allows you to find any term in the sequence without listing out all preceding terms. You use this approach to save time and effort, especially in longer sequences.
The formula for finding the nth term \(a_n\) of a geometric sequence is:\[a_n = a_1 \cdot r^{n-1}\]Where:

• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number you want to find.

For example, to find the fifth term \(a_5\): \[a_5 = 7 \cdot \left( \frac{2}{3} \right)^{4}\]
Calculate the powers and multiply:\[a_5 = 7 \cdot \frac{16}{81} = \frac{112}{81}\]
This simplifies your calculation when working with the sequence. Always use this formula when you specifically need a term, regardless of the sequence's length.

Geometric Sequence Formula

A geometric sequence formula is vital for understanding how sequences behave. The standard formula is:\[ a_n = a_1 \cdot r^{n-1} \]This formula helps determine any term you need within a sequence quickly. Let’s break it down:

• \( a_1 \) is the sequence's first term, which is your start point and a crucial element. For our sequence, it's 7.
• \( r \) is the common ratio, the constant factor between any two successive terms, which is \( \frac{2}{3} \) here.
• \( n \) indicates the specific position of the term in the sequence.

The formula showcases the exponential growth or decay of the sequence, influenced directly by the common ratio.
Applying the formula makes finding the nth term exact and efficient. Whether increasing or diminishing, this formula is your guide to navigate through any geometric sequence challenges you encounter.