Problem 54 - Algebra for College Students

Question

\(4,-\frac{4}{3}, \frac{4}{9},-\frac{4}{27}, \ldots\) 3

Step-by-Step Solution

Answer

The third term is \(\frac{4}{9}\).

Step 1: Identify the pattern

Observe the sequence: \(4, -\frac{4}{3}, \frac{4}{9}, -\frac{4}{27}, \ldots\). Notice that each term is derived by multiplying the previous term by \(-\frac{1}{3}\). This reveals that the sequence is geometric with a common ratio \(r = -\frac{1}{3}\).

Step 2: Determine the form of the sequence

In a geometric sequence, each term can be expressed as \(a_n = a_1 \cdot r^{n-1}\) where \(a_1\) is the first term and \(r\) is the common ratio. Here, \(a_1 = 4\) and \(r = -\frac{1}{3}\).

Step 3: Find the explicit formula

Use the identified first term and common ratio to write the explicit formula for the sequence: \(a_n = 4 \left(-\frac{1}{3}\right)^{n-1}\).

Step 4: Calculate the third term

Substitute \(n = 3\) into the explicit formula: \(a_3 = 4 \left(-\frac{1}{3}\right)^{3-1} = 4 \left(-\frac{1}{3}\right)^2 = 4 \cdot \frac{1}{9} = \frac{4}{9}\).

Key Concepts

Common Ratio

In any geometric sequence, one specific feature that defines it is the **common ratio**. This ratio is the factor by which we multiply each term of the sequence to obtain the next term. It's a constant value throughout the sequence, making it a fundamental aspect.

For example, looking at the sequence from the exercise, which is:

• 4
• -\(\frac{4}{3}\)
• \(\frac{4}{9}\)
• -\(\frac{4}{27}\)

We can identify the common ratio, denoted by \( r \), by dividing any term by the previous one. Doing this calculation:\[ r = \frac{4}{9} \times \left(-\frac{4}{3}\right)^{-1} = -\frac{1}{3} \]This value \(-\frac{1}{3}\) is multiplied by each term to yield the next, confirming the common ratio of the sequence.

Explicit Formula

The **explicit formula** for a geometric sequence is a powerful tool that allows you to compute any term in the sequence without needing to know the preceding ones. This formula is written as:\[ a_n = a_1 \cdot r^{n-1} \]Where:

• \( a_n \) is the term you're trying to find,
• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio,
• \( n \) is the position of the term.

In our example:

• First term \( a_1 = 4 \)
• Common ratio \( r = -\frac{1}{3} \)

Thus, the explicit formula becomes:\[ a_n = 4 \left(-\frac{1}{3}\right)^{n-1} \]This equation can be employed to find any term desired in this sequence.

Sequence Pattern

The **sequence pattern** in a geometric progression is quite straightforward. It follows a systematic approach, unlike an arithmetic sequence, where you add or subtract a constant to get from one term to the next. In a geometric sequence, you always **multiply** by the common ratio.

• Start with the first term.
• Consistently multiply by the common ratio to find successive terms.

For instance, using the original sequence:

• Begin with 4.
• Multiply 4 by \(-\frac{1}{3}\) to get \(-\frac{4}{3}\).
• Then multiply \(-\frac{4}{3}\) by \(-\frac{1}{3}\) to get \(\frac{4}{9}\).
• And so it continues...

This repeating process highlights the geometric sequence's consistency and predictability.

Geometric Progression

A **geometric progression** 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. Such sequences are quite distinct from arithmetic ones, and they exhibit exponential growth or decay because each term is the product of the prior term and a constant ratio.To understand this more deeply, consider:

• The sequence given: \(4, -\frac{4}{3}, \frac{4}{9}, -\frac{4}{27}, ...\)
• The common ratio \( r = -\frac{1}{3} \) indicates a shifting pattern that oscillates signs and diminishes values, showing a decay as terms progress.

Geometric progressions are found not just in theoretical exercises but in real-world phenomena too, such as population growth or radioactive decay, due to their ability to model exponential patterns.