Problem 12 - A Graphical Approach to College Algebra

Question

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$-2,6,-18,54, \dots$$

Step-by-Step Solution

Answer

\(a_5 = -162\), \(a_n = -2 \times (-3)^{n-1}\).

Step 1: Identify the Common Ratio

A geometric sequence is defined by a constant ratio between consecutive terms. Let's find the common ratio \( r \) using the first two terms. The first term \( a_1 = -2 \) and the second term \( a_2 = 6 \). To find \( r \), divide \( a_2 \) by \( a_1 \): \( r = \frac{a_2}{a_1} = \frac{6}{-2} = -3 \).

Step 2: Calculate the Fifth Term

The nth term of a geometric sequence is given by the formula \( a_n = a_1 \times r^{n-1} \). To find \( a_5 \), substitute \( a_1 = -2 \), \( r = -3 \), and \( n = 5 \) into the formula: \[ a_5 = -2 \times (-3)^{5-1} = -2 \times (-3)^{4} = -2 \times 81 = -162 \].

Step 3: Derive the General Formula for \(a_n\)

The nth term formula for a geometric sequence is \( a_n = a_1 \times r^{n-1} \). We already know \( a_1 = -2 \) and \( r = -3 \). Substitute these values into the formula to get: \( a_n = -2 \times (-3)^{n-1} \).

Key Concepts

Common Ratio

A geometric sequence is a series of numbers where each term after the first is produced by multiplying the previous term by a fixed, non-zero number called the *common ratio*.
This is what makes geometric sequences distinct from arithmetic sequences.
To identify the common ratio in any sequence, you simply divide any term (after the first one) by its preceding term.

In the given exercise, the series starts with \(-2\) and proceeds to \(6\).
To find the common ratio \(r\), divide the second term by the first:

• \(a_1 = -2\)
• \(a_2 = 6\)
• The common ratio \(r = \frac{a_2}{a_1} = \frac{6}{-2} = -3\)

This calculation shows that by multiplying each term by \(-3\), you arrive at the next term, continuing the pattern throughout the sequence.

Fifth Term Calculation

Once you have the common ratio \(r\), you can calculate any term in the sequence using the *nth term formula*:
\(a_n = a_1 \times r^{n-1}\).
This formula allows you to find the value of a specific term based on its position in the sequence.

For the fifth term \(a_5\):

• First term \(a_1 = -2\)
• Common ratio \(r = -3\)
• Position \(n = 5\)

Substitute these values into the formula:\[ a_5 = -2 \times (-3)^{5-1} = -2 \times (-3)^{4} = -2 \times 81 = -162 \]
So, the fifth term \(a_5\) is \(-162\).
This calculation demonstrates how exponential growth in a sequence can significantly change the value as you progress through terms.

General Term Formula

The general term formula for a geometric sequence is a powerful tool that allows you to find any term within the sequence.
This formula is composed of the first term \(a_1\), the common ratio \(r\), and the position of the term within the sequence \(n\).

The formula is written as:\(a_n = a_1 \times r^{n-1}\).
This formula neatly encapsulates the pattern of multiplying each term by the common ratio to find the next one.

• Given \(a_1 = -2\)
• Common ratio \(r = -3\)
• The general formula becomes \(a_n = -2 \times (-3)^{n-1}\)

This formula can now be used to calculate any term in the series, whether you're looking for the fifth, tenth, or hundredth term.
Understanding this formula gives you insight into the sequence's behavior as it progresses, showing exponential growth in either the positive or negative direction.