Problem 59 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the geometric sequence. $$a_{2}=-1, a_{7}=-32$$

Step-by-Step Solution

Answer

The general term is \( a_n = -\frac{1}{2} \times 2^{n-1} \).

Step 1: Understand the Problem

We are given a geometric sequence with specific terms, namely \( a_2 = -1 \) and \( a_7 = -32 \). Our task is to find the general term \( a_n \) of this sequence.

Step 2: Recall the Formula for a Geometric Sequence

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, \( r \). The general formula for a geometric sequence is \( a_n = a_1 r^{n-1} \).

Step 3: Use Given Terms to Set Up Equations

We know \( a_2 = a_1 r = -1 \) and \( a_7 = a_1 r^6 = -32 \). These two equations will help us find \( a_1 \) and \( r \).

Step 4: Solve for the Common Ratio \( r \)

Divide the two equations: \( \frac{a_7}{a_2} = \frac{-32}{-1} = 32 \). This results in \( \frac{a_1 r^6}{a_1 r} = r^5 = 32 \) hence \( r = 32^{1/5} = 2 \).

Step 5: Solve for the First Term \( a_1 \)

Substitute \( r = 2 \) back into the equation \( a_1 r = -1 \). This gives \( a_1 \cdot 2 = -1 \), thus \( a_1 = -1/2 \).

Step 6: Write the General Term Formula

Substitute \( a_1 = -1/2 \) and \( r = 2 \) into the general term formula. So the general term \( a_n = a_1 r^{n-1} = -\frac{1}{2} \times 2^{n-1} \).

Key Concepts

Common Ratio

In a geometric sequence, one of the key concepts we encounter is the common ratio, denoted as \( r \). This is the consistent factor by which we multiply one term to get the next term in the sequence. It remains constant throughout the sequence.
Understanding this multiplier is crucial because it allows us to analyze and predict future terms.
In the given problem, we have specific terms of the sequence, \( a_2 = -1 \) and \( a_7 = -32 \). By setting up the ratio of these terms \( \frac{a_7}{a_2} \), we can solve for \( r \). Finding \( r \) involved calculating the fifth root of 32, which simplifies to \( r = 2 \).
To summarize, **the common ratio** is:

• Identified from known terms.
• A constant key to forming other terms.
• Necessitated for the sequence's consistency.

General Term Formula

The general term formula of a geometric sequence is pivotal for determining any term's value in the sequence without needing all preceding terms. It is given by the expression \( a_n = a_1 r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
This formula essentially tells us that any term in the sequence can be calculated if we know the initial term and the common ratio.
In the problem at hand, once we established that \( r = 2 \) and found \( a_1 = -\frac{1}{2} \), we could then use these values in the general term formula. Substituting, we find the general expression to be \( a_n = -\frac{1}{2} \times 2^{n-1} \).
This powerful formula enables easy calculations of terms at any position \( n \). It serves not only for computational effectiveness but also for a better understanding of the behavior of the sequence.

Sequence Analysis

Sequence analysis involves examining and understanding the structure and behavior of the sequence. Analyzing a geometric sequence requires understanding both the arrangement and progression of its terms.
For the given sequence, we've uncovered the common ratio \( r = 2 \) and the first term \( a_1 = -\frac{1}{2} \). This shows a consistent pattern of growth, where each term is twice the previous one while starting from \( -\frac{1}{2} \).
Such analysis helps us predict and describe how each term grows over each iteration. It is especially useful for:

• Predicting future terms' values.
• Understanding the sequence’s overarching behavior — here, a doubling effect.
• Identifying properties like limits or common factors.

Ultimately, **sequence analysis** underscores not just the mechanics but also the elegance and simplicity behind the progression of each term.