Problem 28 - College Algebra

Question

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{0.61,1.83,5.49,16.47, \ldots\\} $$

Step-by-Step Solution

Answer

The recursive formula is \( a_{n} = 3 \, a_{n-1} \) with \( a_{1} = 0.61 \).

Step 1: Identifying the Pattern

First, identify the consistent multiplier, or common ratio, between consecutive terms. To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{1.83}{0.61} = 3 \). Verify the ratio by checking subsequent terms: \( \frac{5.49}{1.83} = 3 \) and \( \frac{16.47}{5.49} = 3 \). Thus, the common ratio \( r \) is 3.

Step 2: Writing the Recursive Formula

Using the initial term and the common ratio identified, we can write the recursive formula for the sequence. A recursive formula for a geometric sequence has the form \( a_{n} = r \, a_{n-1} \) with \( a_{1} \) given. Since \( a_{1} = 0.61 \) and \( r = 3 \), the recursive formula is \( a_{n} = 3 \, a_{n-1} \) with \( a_{1} = 0.61 \).

Key Concepts

Recursive Formula

A recursive formula is an equation that expresses each term in a sequence using the preceding terms. It provides a way to generate the entire sequence by repeatedly applying the formula, instead of having to compute each term independently. In the given problem, the task was to find a recursive formula for a particular geometric sequence.For a geometric sequence, the recursive formula includes:

• The common ratio, which determines how each term relates to its predecessor,
• The initial term, which is the starting point of the sequence.

For example, given the sequence starting with 0.61, with each term multiplied by a common ratio of 3, the recursive formula is expressed as:\[a_{n} = 3 \, a_{n-1}\]This formula allows us to find any term by knowing the one before it, connecting the whole sequence back to its initial value.

Common Ratio

The common ratio is a key component of a geometric sequence. It represents the factor by which we multiply each term to get the next term. Finding the common ratio is the first step in writing a recursive formula.To determine the common ratio in this example, we divide a term in the sequence by its previous term. Starting with the second term:\[r = \frac{1.83}{0.61} = 3\]We verify this by checking other consecutive terms:

• \(\frac{5.49}{1.83} = 3\)
• \(\frac{16.47}{5.49} = 3\)

The consistency of these results confirms that the common ratio is indeed 3. This ratio is crucial as it dictates how the sequence evolves, and is an essential part of the recursive formula.

Sequence Pattern

Understanding the sequence pattern in a geometric series helps in finding the recursive formula. A sequence pattern in a geometric sequence is primarily defined by its common ratio. As each term is derived by multiplying the previous one by this ratio, a predictable pattern emerges. In our geometric sequence:

• Start with 0.61,
• Multiply by 3 to get the next term.

This repeating multiplication forms a clear pattern:

• 0.61,
• 1.83,
• 5.49,
• 16.47,
• ...and so on.

By grasping this sequence pattern, it's easier to see the logic behind the recursive formula. It assures that any disruption in the common ratio will alter the overall pattern.

College Algebra

In college algebra, sequences like the one in this exercise are fundamental concepts. They serve as building blocks for understanding more complex algebraic structures and relationships. The ability to work with recursive formulas and geometric sequences is crucial. It enables students to tackle more complex mathematical problems, as these concepts often appear in various algebraic and real-world contexts. Learning to find the common ratio and recognize sequence patterns develops valuable problem-solving skills. These skills help students understand how to apply algebraic techniques to find solutions efficiently. With practice, these algebraic concepts become intuitive, fostering a deeper appreciation for the structure and predictability of mathematics.