Problem 24 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th 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 \(n\)th term is \(a_n = 7 \cdot \left(\frac{2}{3}\right)^{n-1}\).

Step 1: Identify the first term

The first term of the sequence is given as \(a_1 = 7\).

Step 2: Find the common ratio

To find the common ratio \(r\) of a geometric sequence, divide the second term by the first term: \(r = \frac{a_2}{a_1} = \frac{\frac{14}{3}}{7} = \frac{2}{3}\).

Step 3: Calculate the fifth term

The formula for the \(n\)th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\). To find the fifth term, substitute \(n = 5\): \(a_5 = 7 \cdot \left(\frac{2}{3}\right)^{4} = 7 \cdot \frac{16}{81} = \frac{112}{81}\).

Step 4: Write the formula for the nth term

Using the formula for the \(n\)th term of a geometric sequence, with \(a_1 = 7\) and \(r = \frac{2}{3}\), we have: \(a_n = 7 \cdot \left(\frac{2}{3}\right)^{n-1}\).

Key Concepts

Understanding the Common Ratio

A geometric sequence is characterized by its constant multiplier known as the 'common ratio'. To find this ratio, we divide any term in the sequence by the preceding term. In the given sequence, the common ratio, denoted as \(r\), is determined by dividing the second term by the first term.
So, for the sequence \(7, \frac{14}{3}, \frac{28}{9}, \frac{56}{27}, \dots\), the common ratio \(r\) is:

• Divide the second term by the first: \( \frac{\frac{14}{3}}{7} \)
• Simplify the fraction: \( \frac{2}{3} \)

This common ratio \( \frac{2}{3} \) tells us how each term is related to the next. Each term in the sequence is \( \frac{2}{3} \) times the term before it. This consistent ratio allows us to predict other terms in the sequence easily.

Calculating the nth Term of a Geometric Sequence

To find any specific term in a geometric sequence, we use the formula for the \(n\)th term, written as \(a_n = a_1 \cdot r^{n-1}\). This formula allows us to plug in the values of the first term \(a_1\), the common ratio \(r\), and the position \(n\) of the term we want to find.
For our sequence with a first term \(a_1 = 7\) and a common ratio \(r = \frac{2}{3}\), the formula becomes:

• \(a_n = 7 \cdot \left(\frac{2}{3}\right)^{n-1}\)

Here’s a simple way to understand why this formula works:

• The exponent \(n-1\) indicates how many times we multiply by the common ratio, starting from the first term.
• If \(n = 1\), we don’t multiply by the ratio, so we only have the first term, \(a_1\).
• For \(n = 2\), we multiply the first term once by the common ratio, and so forth.

This formula gives us a powerful tool to find whatever term we need in a geometric sequence.

Finding the Fifth Term

To determine the fifth term of the geometric sequence, apply the formula for the \(n\)th term where \(n = 5\). With our sequence, we have:

• The first term \(a_1 = 7\)
• The common ratio \(r = \frac{2}{3}\)

Plug these values into the formula \(a_n = a_1 \cdot r^{n-1}\):

• Calculate the fifth term: \(a_5 = 7 \cdot \left(\frac{2}{3}\right)^{4}\)
• This simplifies to \(a_5 = 7 \cdot \frac{16}{81} = \frac{112}{81}\)

The multiplication gives us the precise fifth term, \(\frac{112}{81}\).
Understanding the process to get to this term demonstrates the beauty of patterns in geometric sequences and the power of mathematical formulas to uncover them. Each subsequent application of the common ratio builds upon the previous steps, reflecting the exponential growth characteristic of such sequences.