Problem 28 - Precalculus Mathematics for Calculus

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 formula for the \( n \)-th term is \( 7 \times \left( \frac{2}{3} \right)^{n-1} \).

Step 1: Identify the Common Ratio

To find the common ratio, divide the second term by the first term of the sequence. Here, the second term is \( \frac{14}{3} \) and the first term is \( 7 \). Calculate \( \frac{14/3}{7} \).

Step 2: Calculate the Common Ratio

Simplify \( \frac{14/3}{7} = \frac{14/3} \times \frac{1}{7} = \frac{14}{21} = \frac{2}{3} \). Thus, the common ratio \( r \) is \( \frac{2}{3} \).

Step 3: Find the Fifth Term

The formula for the \( n \)-th term of a geometric sequence is \( a_n = a_1 \times r^{n-1} \). Here, \( a_1 = 7 \), and \( r = \frac{2}{3} \). Substitute \( n = 5 \): \( a_5 = 7 \times \left( \frac{2}{3} \right)^{4} \).

Step 4: Calculate the Fifth Term

Compute the expression: \( a_5 = 7 \times \left( \frac{2}{3} \right)^{4} = 7 \times \frac{16}{81} = \frac{112}{81} \). Thus, the fifth term is \( \frac{112}{81} \).

Step 5: Derive the General Formula for the n-th Term

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

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key concept used to determine how each term relates to the previous one. It is the constant factor you multiply by to go from one term to the next. To find this, take any term in the sequence, except the first one, and divide it by its preceding term. This ensures you have a consistent ratio throughout the sequence.

For example, in the sequence provided: \(7, \frac{14}{3}, \frac{28}{9}, \frac{56}{27}, \dots\), we find the common ratio \(r\) by dividing the second term \(\frac{14}{3}\) by the first term \(7\). This simplifies to \(\frac{14/3}{7} = \frac{2}{3}\).

It’s helpful to note:

• The formula to find the common ratio is \( r = \frac{a_2}{a_1} \).
• A positive common ratio indicates terms increase or decrease positively, while a negative ratio flips signs between terms.

Fifth Term Calculation

Calculating a specific term in a geometric sequence, like the fifth term, involves using the common ratio and the first term of the sequence. The general formula to find any term \(a_n\) is: \[ a_n = a_1 \times r^{(n-1)} \]

Here:

• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio found previously.
• \(n\) is the position of the term in the sequence we want to find.

Using the sequence details: the fifth term \(a_5\) is calculated by substituting the values \(a_1 = 7\), \(r = \frac{2}{3}\), and \(n = 5\) into the formula:
\[ a_5 = 7 \times \left( \frac{2}{3} \right)^4 \] Simplifying, this becomes \(7 \times \frac{16}{81} = \frac{112}{81}\). Therefore, the fifth term of the sequence is \(\frac{112}{81}\).

Knowing how to compute specific terms is crucial, as it allows you to explore the properties of infinite sequences and understand their behavior over time.

n-th Term Formula

The n-th term formula of a geometric sequence provides a way to calculate the positionally known term without listing all the preceding terms. This is particularly useful for large sequences where it wouldn't be practical to calculate every term.

In general, the formula is:\[a_n = a_1 \times r^{(n-1)}\]

Where:

• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the term number.

Using this formula and the given sequence starting with \(a_1 = 7\) and a common ratio of \(\frac{2}{3}\), the expression becomes:\[ a_n = 7 \times \left( \frac{2}{3} \right)^{n-1} \] This formula allows you to find any term in the sequence without manual computations of every intermediate step.

For instance, if you wanted the 10th term, simply plug in \(n = 10\) to get:\[ a_{10} = 7 \times \left( \frac{2}{3} \right)^9 \] This power and multiplication give you the term you are looking for. With this formula, understanding and analyzing sequences becomes more manageable.