Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$4,1.2,0.36,0.108, \ldots$$

Step-by-Step Solution

Answer

5th term: 0.0324; 8th term: 0.0008748; nth term: \(a_n = 4 \cdot (0.3)^{n-1}\).

Step 1: Identify the first term and common ratio

In a geometric sequence, the first term is denoted by \(a\). Here, \(a = 4\). To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{1.2}{4} = 0.3\). Thus, the common ratio \(r = 0.3\).

Step 2: Use the formula for the nth term of a geometric sequence

The formula for the \(n\)th term \(a_n\) in a geometric sequence is \(a_n = a \cdot r^{n-1}\). Here, \(a = 4\), and \(r = 0.3\).

Step 3: Find the 5th term

To find the 5th term, set \(n = 5\) in the nth term formula: \(a_5 = 4 \cdot (0.3)^{5-1} = 4 \cdot (0.3)^4\). Calculate \((0.3)^4 = 0.0081\), so \(a_5 = 4 \cdot 0.0081 = 0.0324\).

Step 4: Find the 8th term

To find the 8th term, set \(n = 8\): \(a_8 = 4 \cdot (0.3)^{8-1} = 4 \cdot (0.3)^7\). Calculate \((0.3)^7 = 0.0002187\), and thus \(a_8 = 4 \cdot 0.0002187 = 0.0008748\).

Step 5: Define the nth term expression

The expression for the \(n\)th term is \(a_n = 4 \cdot (0.3)^{n-1}\). This formula can be used to find any term in the sequence by substituting for \(n\).

Key Concepts

Understanding the nth Term Formula

In a geometric sequence, each term is derived by multiplying the previous term by a constant, known as the common ratio. To find any term in a geometric sequence without listing all the terms, we use the nth term formula. This formula is:

• \( a_n = a \cdot r^{n-1} \)

Here, \( a \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence. By using this formula, we can easily find any term in a sequence, even if it is very far along. The formula saves time, especially for sequences with a large number of terms.
This expression helps to pinpoint any desired term, making calculations faster and more efficient, without needing to compute all preceding terms consecutively.

Deciphering the Common Ratio

The common ratio is fundamental in a geometric sequence as it dictates the relationship between consecutive terms. It is calculated by dividing any term in the sequence by the preceding term. For instance, in our earlier example, the common ratio \( r \) is derived as follows:

• \( r = \frac{1.2}{4} = 0.3 \)

The common ratio is a crucial component as it determines the nature of the sequence:

• When \( r \) is greater than 1, the sequence is expanding or increasing.
• If \( 0 < r < 1 \), like in our exercise, the sequence is decreasing.
• A negative ratio results in alternating positive and negative terms.

Understanding the common ratio helps establish the pattern of the sequence, aiding in predicting further terms.

Exploring Geometric Sequences in Mathematics

Geometric sequences are an essential type of numerical sequence in mathematics, known for their properties and applications. In a geometric sequence, each term is obtained by multiplying the previous term by a consistent factor, termed the common ratio. This characteristic allows geometric sequences to model phenomena in science, finance, and nature efficiently.
Some key points about geometric sequences include:

• The power of the common ratio reflects the term's position, emphasizing the exponential nature of these sequences.
• Geometric sequences may progress towards zero, infinity, or oscillate depending on the common ratio.
• They play a vital role in modeling exponential growth or decay, such as calculating compound interest or population growth.

Mastering geometric sequences offers valuable insight into recognizing and solving problems involving exponential patterns in various fields.