Problem 5 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{1}=5, r=-2$$

Step-by-Step Solution

Answer

\(a_5 = 80\), \(a_n = 5 \, (-2)^{n-1}\).

Step 1: Understanding a Geometric Sequence

First, recognize that a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (\(r\)). In this problem, the first term \(a_1 = 5\) and the common ratio \(r = -2\).

Step 2: Finding the General Formula for the n-th Term

The formula for the n-th term of a geometric sequence is given by \(a_n = a_1 \, r^{n-1}\). This formula allows us to find any term in the sequence once we know the first term and the common ratio.

Step 3: Calculating the 5th Term \(a_5\)

To find the 5th term, set \(n = 5\) in the formula: \[ a_5 = 5 \times (-2)^{5-1} = 5 \times (-2)^4. \] Calculate \((-2)^4\), which is \(16\), and then multiply by 5: \[ a_5 = 5 \times 16 = 80. \]

Step 4: Expressing the General Formula \(a_n\)

Substitute \(a_1 = 5\) and \(r = -2\) into the general formula: \[ a_n = 5 \, (-2)^{n-1}. \] This expression represents the n-th term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element that determines how each term relates to the previous one. It is calculated by dividing any term in the sequence by its immediate predecessor, except the first term.
The common ratio can be either positive or negative, and this sign greatly impacts the sequence's behavior.

• A positive common ratio means all terms are either all positive or all negative, gradually growing or shrinking based on the ratio's absolute value.
• A negative common ratio results in terms that alternate signs. This alternating pattern can rapidly change the sequence's direction if the ratio is less than -1 or gently bounce around when between -1 and 0.

Understanding the impact of the common ratio helps predict the sequence's behavior and spot any patterns or anomalies.

n-th Term Formula

The n-th term formula is your go-to tool for finding any term in a geometric sequence without listing all previous terms. This formula is given by \[a_n = a_1 \, r^{n-1}\]where:

• \(a_n\) is the n-th term you're looking to find,
• \(a_1\) is the first term in the sequence, and
• \(r\) is the common ratio.

This formula helps you leap directly to your desired term by only needing the initial term and the common ratio.
For example, in our given sequence with \(a_1 = 5\) and \(r = -2\), to find any term like the 5th term, you'd set \(n = 5\) in the formula to calculate \(a_5\), making this approach efficient for any term number.

Sequence

A sequence is a set of numbers arranged in a specific order following a particular rule. A geometric sequence, the type we're focusing on, has a pattern based on multiplication.
From the first term and the common ratio, you can generate the entire sequence by repeatedly multiplying by the common ratio.

• The sequence starts with an initial term, often designated as \(a_1\).
• Each subsequent term is found by continuing to multiply by the common ratio.

Let's look at an example: Starting with \(a_1 = 5\) and a common ratio of \(-2\), the sequence progresses as follows: \[5, -10, 20, -40, 80, \ldots\] The alternating signs are due to the negative common ratio, highlighting the importance of understanding how sign and magnitude influence the sequence's development.