Problem 28 - College Algebra with Corequisite Support

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 \(a_1 = 0.61\), \(a_n = a_{n-1} \cdot 3.00\).

Step 1: Identify the First Term

The first term of the geometric sequence is the initial value, which is given as \(a_1 = 0.61\).

Step 2: Determine the Common Ratio

Find the common ratio \(r\) by dividing the second term by the first term. Calculate \( r = \frac{1.83}{0.61} \approx 3.00 \). This process shows how the sequence increases from one term to the next.

Step 3: Write the Recursive Formula

Using the first term and common ratio, the recursive formula for a geometric sequence is expressed as follows: \(a_1 = 0.61\), and the general recursive formula is \(a_n = a_{n-1} \cdot r\). Substitute the common ratio: \(a_n = a_{n-1} \cdot 3.00\).

Key Concepts

Geometric Sequence

A geometric sequence 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. This type of sequence is very systematic which makes it predictable. Each term is formed by using the previous term, making it a great way to understand repeated multiplication. In the example given in the exercise, you started with the first term, 0.61, and continue multiplying by the common ratio, which you discovered to be approximately 3.00. This repetition forms the entire sequence. Geometric sequences are everywhere, from population growth to financial forecasts.

• Every term is calculated by multiplying the previous term by the common ratio.
• The sequence either increases or decreases exponentially depending on the common ratio.

Recognizing this pattern is key to working not only with recursive formulas but also seeing possibilities of real-world applications.

Common Ratio

The common ratio is the backbone of a geometric sequence. It is what connects one term to the next. To find this ratio, you simply divide any term by its previous term. The value remains constant throughout the sequence, signifying the amount of growth or reduction between consecutive terms. For example, in our exercise, dividing the second term (1.83) by the first term (0.61) resulted in approximately 3.00. This constant ratio shows how the sequence progresses. The concept of the common ratio is crucial because:

• It dictates the rate at which the terms increase or decrease.
• A positive ratio greater than 1 indicates growth, whereas a ratio between 0 and 1 indicates a decrease.
• If the common ratio is negative, the terms will alternate between positive and negative, adding another layer of complexity to the sequence.

Understanding the common ratio equips you with the ability to navigate through the sequence and foresee upcoming values.

First Term

The journey of every arithmetic or geometric sequence begins with the first term. This term serves as the starting point from which all other parts of the sequence develop. In geometric sequences, the first term, often denoted as \(a_1\), is crucial for setting up the pattern presented in the sequence. In this particular exercise, the first term is given as 0.61.Here’s why the first term matters:

• It establishes the base from which progression occurs.
• Combined with the common ratio, it allows you to formulate the recursive formula that defines the entire sequence.
• Any change in the first term significantly alters the entire sequence's values while keeping the form the same.

With the first term defined, one can build the entire sequence by repeatedly applying the common ratio.