Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$162,-54,18,-6, \ldots$$

Step-by-Step Solution

Answer

5th term is 2, 8th term is \(-\frac{2}{27}\).

Step 1: Identify the first term

The first term of the sequence is the initial term given, which is 162. This is denoted as \(a_1 = 162\).

Step 2: Find the common ratio

To find the common ratio \(r\), divide the second term by the first term: \(-54 \div 162 = -\frac{1}{3}\). Thus, \(r = -\frac{1}{3}\).

Step 3: General formula for the nth term

The formula for the nth term of a geometric sequence is given by \(a_n = a_1 \, r^{n-1}\). Here, \(a_1 = 162\) and \(r = -\frac{1}{3}\).

Step 4: Calculate the nth term formula

Substitute \(a_1\) and \(r\) into the formula: \(a_n = 162 \, \left(-\frac{1}{3}\right)^{n-1}\). This will give us the nth term for any \(n\).

Step 5: Calculate the fifth term

Substitute \(n = 5\) into the nth term formula: \[a_5 = 162 \, \left(-\frac{1}{3}\right)^{5-1} = 162 \, \left(-\frac{1}{3}\right)^4 = 162 \, \times \frac{1}{81} = 2\].

Step 6: Calculate the eighth term

Substitute \(n = 8\) into the nth term formula: \[a_8 = 162 \, \left(-\frac{1}{3}\right)^{8-1} = 162 \, \left(-\frac{1}{3}\right)^7\].Calculate \((-\frac{1}{3})^7 = -\frac{1}{2187}\), hence \[a_8 = 162 \, \times -\frac{1}{2187} = -\frac{162}{2187} = -\frac{2}{27}\].

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial component. It's the factor by which we multiply one term to obtain the next term in the sequence. To find this ratio, you can simply divide the second term by the first term of the sequence.
For example, if our sequence begins with 162 and the second term is -54, we'd find the common ratio like this:

• Divide -54 by 162, which gives you \(-\frac{1}{3}\).

This means that each term in the sequence is produced by multiplying the previous term by -\(\frac{1}{3}\). It's important to remember that the common ratio can be positive or negative, and as seen here, it can also be a fraction. The sign and magnitude of the common ratio affect the direction and rate of growth or shrinkage of the sequence.

nth Term Formula

The ability to find any term in a geometric sequence relies on a straightforward formula. This formula is structured to make predictions easy once you know the sequence's first term and its common ratio. To find the \( n \)th term, use this formula:

• \( a_n = a_1 \cdot r^{n-1} \)

Where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you wish to find.
Suppose we have a sequence where the first term \( a_1 \) is 162 and the common ratio \( r \) is -\(\frac{1}{3}\). If we use the formula to find the 5th term, we substitute \( n = 5 \) into the formula:

• \( a_5 = 162 \cdot (-\frac{1}{3})^{5-1} = 2 \)

The nth term formula allows for quick calculations for any position \( n \), providing a powerful tool for analyzing geometric sequences.

Geometric Progression

A geometric progression is essentially a list of numbers in which each term after the first is found by multiplying the previous one by the common ratio. This type of sequence can appear everywhere—from investments with constant interest rates to biological systems with exponential growth. Understanding the geometric progression:

• The sequence starts with a starting value or first term \( a_1 \).
• Each subsequent term \( a_n \) is the product of the preceding term and the common ratio \( r \).

For instance, in the sequence \(162, -54, 18, -6, \ldots\), you can observe the geometric progression as each term is generated by multiplying the previous term by the common ratio \(-\frac{1}{3}\).
The beauty of geometric progressions is their predictability and simplicity once you understand the basic pattern. Detecting such structures in datasets or patterns helps in forecasting and making informed decisions in various fields.