Problem 10 - Precalculus: Mathematics for Calculus

Question

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=-3, \quad r=-2$$

Step-by-Step Solution

Answer

The fourth term is 24.

Step 1: Identify the General Formula

The nth term of a geometric sequence is given by the formula: \( a_n = a \cdot r^{n-1} \). Where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Plug in Given Values

For the given problem, the first term \(a = -3\) and the common ratio \(r = -2\). Therefore, the formula for the nth term becomes: \( a_n = -3 \cdot (-2)^{n-1} \).

Step 3: Calculate the Fourth Term

To find the fourth term, substitute \(n = 4\) into the formula: \( a_4 = -3 \cdot (-2)^{4-1} = -3 \cdot (-2)^3 \).

Step 4: Simplify the Expression

Calculate \((-2)^3\). Since \((-2)^3 = -8\), substitute this back into the expression: \( a_4 = -3 \cdot (-8) \).

Step 5: Final Calculation

Now perform the multiplication: \(-3 \cdot (-8) = 24\). Thus, the fourth term of the sequence is 24.

Key Concepts

nth term

When we talk about a geometric sequence, each term is generated through a formula that depends on its position, or "n" value, within the series. The formula for the nth term is expressed as: \[ a_n = a \cdot r^{n-1} \]Where:

• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• n is the position of the term within the sequence.

This formula lets you compute any term in the sequence by substituting the appropriate values for \(a\), \(r\), and \(n\). For instance, in our exercise:The given first term \(a\) is \(-3\), the common ratio \(r\) is \(-2\), so to find the nth term formula for this sequence, it becomes: \[ a_n = -3 \cdot (-2)^{n-1} \].This way, you can find any term you want without having to go through each preceding term.

common ratio

The common ratio in a geometric sequence is the key element that differentiates it from an arithmetic sequence. It is the factor by which we multiply each term to get to the next one.

• To find the common ratio \(r\), you can divide any term in the sequence by its preceding term.
• The formula for the common ratio is:\[ r = \frac{a_n}{a_{n-1}} \]

For instance, in our example, the common ratio is \(-2\). It means each term is the result of multiplying the previous term by \(-2\). With this in mind, let's see how it works:

• If the first term is \(-3\), then the second term is \(-3 \times (-2) = 6\).
• The third term becomes \(6 \times (-2) = -12\).

Understanding the common ratio is crucial because it defines the flow and pattern of a geometric sequence.

geometric progression

A geometric progression—or geometric sequence—is a mathematical sequence in which each term is obtained by multiplying the previous one by a fixed, non-zero number known as the common ratio. Here’s what makes it stand out:

• Each step in the progression maintains a consistent pattern of multiplication by the common ratio.
• The sequence can quickly escalate, or shrink, exponentially depending on whether the common ratio is greater than or less than 1 (or negative).

The general form of a geometric progression starting from the first term \(a\) is:\[ a, \, ar, \, ar^2, \, ar^3, \ldots \]In our sequence example, it starts with \(-3\), and continues \(6, \ -12, \ 24, \ldots\), due to the common ratio of \(-2\). This pattern helps us find any term in the sequence since it follows a key multiplication structure applicable universally throughout the progression.