Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each geometric sequence. \(a_{n}=\left\\{\frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360}, \ldots\right\\}\)

Step-by-Step Solution

Answer

The recursive formula is \(a_1 = \frac{3}{5}\); \(a_n = a_{n-1} \times \frac{1}{6}\) for \(n \geq 2\).

Step 1: Understanding Geometric Sequences

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant known as the 'common ratio'.

Step 2: Identifying the First Term

The first term of the sequence is \(a_1 = \frac{3}{5}\).

Step 3: Finding the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{1}{10} \div \frac{3}{5} = \frac{1}{10} \times \frac{5}{3} = \frac{1}{6}\).

Step 4: Writing the Recursive Formula

The recursive formula for a geometric sequence can be written as \(a_{n} = a_{n-1} \times r\). With \(a_1 = \frac{3}{5}\) and \(r = \frac{1}{6}\), the recursive formula becomes \(a_{n} = a_{n-1} \times \frac{1}{6}\).

Step 5: Complete Recursive Formula

Combine the initial condition and the recursive formula: \(a_1 = \frac{3}{5}\); \(a_n = a_{n-1} \times \frac{1}{6}\) for \(n \geq 2\).

Key Concepts

Recursive Formula

A geometric sequence is a series of numbers where each term after the first is derived by multiplying the preceding term by a fixed, non-zero number known as the "common ratio." This might sound complex, but it's simply about repeating a pattern consistently. To describe such a series mathematically in a concise way, we use something called a recursive formula.
A recursive formula is a tool that defines how any term in the sequence relates to its previous term. More specifically, in a geometric sequence, the recursive formula looks like this:

• Let the first term be denoted as \(a_1\).
• For the subsequent terms, \(a_n\) is obtained by multiplying the previous term \(a_{n-1}\) by the common ratio \(r\), represented as \(a_n = a_{n-1} \times r\).

This formula helps us quickly find any term in the sequence, making it a very handy method for algebra and college-level math studies.

Common Ratio

In the world of geometric sequences, the 'common ratio' is a key player. But what is it exactly? It's simply the factor that you multiply by to get from one term to the next. It stays constant throughout the series. To figure out the common ratio, take any term in the sequence and divide it by its preceding term.
Let's look at an example to understand this better. Consider a sequence starting with \(\frac{3}{5}\), followed by \(\frac{1}{10}\). To find the common ratio \(r\), calculate:

• \(r = \frac{1}{10} \div \frac{3}{5} = \frac{1}{10} \times \frac{5}{3} = \frac{1}{6}\)

This tells us that each term is derived by multiplying the previous term by \(\frac{1}{6}\). Recognizing this ratio is crucial for forming a recursive formula and understanding the sequence's behavior. In general, a positive common ratio leads to a sequence that grows, while a negative or fractional ratio results in a sequence that decreases or alternates.

College Algebra

In college algebra, one of the essential concepts is understanding sequences and series, particularly geometric sequences. Mastering these not only helps in solving algebraic problems but also builds a foundation for calculus and advanced mathematical studies. This topic starts with recognizing the type of sequence and using formulas to find unknown terms.
A big part of mastering college algebra is working with the recursive formulas we discussed earlier. These are not just theoretical constructs but practical tools for solving real-world problems. For instance, understanding geometric sequences is vital for exploring topics such as exponential growth and decay in fields like finance and the natural sciences.
Another important aspect in college algebra is learning to manipulate expressions and equations accurately. Handling sequences requires a solid grasp of basic arithmetic and algebraic manipulations, like simplifying fractions, which are fundamental skills in this area of study. Building proficiency in these skills makes tackling more advanced topics smoother and more intuitive.