Problem 32 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ -8,-2,-\frac{1}{2},-\frac{1}{8}, \ldots $$

Step-by-Step Solution

Answer

The common ratio is \(\frac{1}{4}\), the fifth term is \(-\frac{1}{32}\), and the nth term is \(a_n = -8 \times \left(\frac{1}{4}\right)^{n-1}\)."

Step 1: Identify the First Term and the Common Ratio

In a geometric sequence, the first term is denoted by \( a_1 \). Here, the first term \( a_1 = -8 \). The common ratio \( r \) is the factor by which we multiply one term to get the next. We find \( r \) by dividing the second term by the first: \( r = \frac{-2}{-8} = \frac{1}{4} \).

Step 2: Find the Fifth Term of the Sequence

To find the fifth term, use the formula for the nth term of a geometric sequence: \( a_n = a_1 \times r^{n-1} \). Plug in the values: \( a_5 = -8 \times \left(\frac{1}{4}\right)^{4} \). Calculate \( \left(\frac{1}{4}\right)^4 = \frac{1}{256} \). Then, \( a_5 = -8 \times \frac{1}{256} = -\frac{1}{32} \).

Step 3: Derive the General nth Term Formula

The formula for the nth term of a geometric sequence is given by \( a_n = a_1 \times r^{n-1} \). Substitute the known values for \( a_1 \) and \( r \): \( a_n = -8 \times \left(\frac{1}{4}\right)^{n-1} \). This is the expression for the nth term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element. It determines how the terms in the sequence change from one to the next. The common ratio, denoted as \( r \), is found by dividing any term in the sequence by its preceding term. For the sequence \(-8, -2, -\frac{1}{2}, -\frac{1}{8}, \ldots\), the common ratio can be calculated by taking the second term \(-2\) and dividing it by the first term \(-8\): \[ r = \frac{-2}{-8} = \frac{1}{4} \]

• This ratio of \( \frac{1}{4} \) tells us that each term in the sequence is obtained by multiplying the previous term by \( \frac{1}{4} \).
• Understanding the common ratio is essential because it provides the multiplicative factor that allows progression through the sequence.

Hence, the consistent application of this ratio between terms creates the pattern unique to geometric sequences.

nth Term Formula

The nth term formula provides a way to find any term in a geometric sequence without having to write out all the preceding terms. The formula for the nth term, \( a_n \), of a geometric sequence is:\[ a_n = a_1 \times r^{n-1} \]

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

For example, to find the fifth term (\( n = 5 \)) of the given sequence, you substitute the values \( a_1 = -8 \) and \( r = \frac{1}{4} \) using the formula: \[ a_5 = -8 \times \left(\frac{1}{4}\right)^{4} \] Calculating, \( \left(\frac{1}{4}\right)^{4} = \frac{1}{256} \) and \( a_5 = -8 \times \frac{1}{256} = -\frac{1}{32} \). This demonstrates how the formula can provide any term directly by utilizing the power of the common ratio and initial term.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is the product of the previous term and a fixed, non-zero number known as the common ratio. The sequence \(-8, -2, -\frac{1}{2}, -\frac{1}{8}, \ldots\) is an example of geometric progression, as:

• Each term is derived by multiplying the previous one by the common ratio \( \frac{1}{4} \).
• It maintains a consistent multiplicative pattern throughout the sequence.

A geometric progression can exhibit either exponential growth or decay based on whether the common ratio is greater than or less than one, respectively. In this particular sequence, the common ratio \( \frac{1}{4} \) is less than one, indicative of exponential decay, where each subsequent term becomes smaller.This simple yet powerful property of geometric sequences is applied in various fields, from calculating population growth and investments to performing computer science algorithms.