Problem 38 - College Algebra

Question

For the following exercises, write a recursive formula for each sequence. $$ 15,3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \ldots $$

Step-by-Step Solution

Answer

The recursive formula is \(a_n = a_{n-1} \cdot \frac{1}{5}\) with \(a_1 = 15\).

Step 1: Identify the Pattern

First, observe the given sequence: \(15, 3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \ldots\). Notice how each term is found by multiplying the previous term by \(\frac{1}{5}\). For example, \(15 \cdot \frac{1}{5} = 3\) and \(3 \cdot \frac{1}{5} = \frac{3}{5}\).

Step 2: Define the Recursive Formula

We know that a recursive formula is expressed as \(a_n = a_{n-1} \cdot r\), where \(r\) is the common ratio. Since the common ratio is \(\frac{1}{5}\), the recursive formula is \(a_n = a_{n-1} \cdot \frac{1}{5}\).

Step 3: Identify the First Term

The first term of the sequence is \(a_1 = 15\). This is important as it serves as the base case for the recursive formula.

Key Concepts

Arithmetic Sequences

Arithmetic sequences are fascinating mathematical arrangements where each term progresses from the previous one by adding a constant value. This constant value is known as the "common difference." Imagine a simple sequence like 2, 4, 6, 8, and so on. Here, each term increases by 2, which is the common difference.

• An arithmetic sequence can be represented as: \(a_n = a_1 + (n-1) imes d\), where "\(a_1\)" is the first term and "\(d\)" is the common difference.
• These sequences are linear, meaning that when plotted on a graph, they form a straight line. This characteristic makes arithmetic sequences predictable and easy to understand.

Despite their simplicity, mistakes can happen when identifying the common difference, especially when working under time pressure or with large numbers. Practice identifying and calculating the common difference to become more proficient in mastering arithmetic sequences.

Geometric Sequences

Geometric sequences present a different and exciting pattern from arithmetic sequences. In a geometric sequence, each term is derived by multiplying the previous term by a fixed, non-zero number known as the "common ratio." For instance, consider the sequence 3, 6, 12, 24, where each term grows by multiplying the previous term by 2.

• A geometric sequence is expressed by the formula: \(a_n = a_1 \times r^{n-1}\), where "\(a_1\)" is the first term and "\(r\)" is the common ratio.
• Unlike arithmetic sequences, geometric sequences increase or decrease exponentially rather than linearly.

These sequences are powerful tools in mathematical modeling, particularly in areas involving exponential growth or decay, such as compound interest calculations or population dynamics. Understanding geometric sequences opens the door to exploring many real-world applications.

Common Ratio

The common ratio in a geometric sequence is the backbone that connects each term to the next. Defined as the constant multiplier between consecutive terms, the common ratio can significantly affect the behavior of the sequence.

• For example, a common ratio greater than 1 results in a sequence that grows, such as 2, 6, 18 (with a common ratio of 3).
• If the common ratio is between 0 and 1, as in the sequence 15, 3, \(\frac{3}{5}\), the sequence decreases, approaching zero.
• The common ratio also determines the nature of summations within infinite sequences, influencing the convergence or divergence of a series.

Knowing how to find and interpret the common ratio is essential when constructing a recursive formula, as demonstrated in the original sequence problem, ensuring that you can seamlessly transition between each term of the sequence with accuracy.