Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 0.3+0.03+0.003+0.0003+\cdots+0.000000003 $$

Step-by-Step Solution

Answer

Yes, the sum is a geometric series. The first term \(a\) is '0.3', and the common ratio \(r\) is '0.1'.

Step 1: Identify if the Sum is Geometric

To identify if the series is geometric, let's check if there is a common ratio, 'r', between consecutive terms. That is, if each term is the product of the previous term and a fixed number. In this case, let's divide the second term (0.03) by the first term (0.3). The result equals 0.1. If we repeat this for the next pair of terms and for the remaining ones, we will see that the quotient remains the same, which means that a constant ratio exists, therefore, the series is geometric.

Step 2: Identify 'a' and 'r'

For a geometric series, the first term 'a' is the first item in the series. In this case, 'a' is 0.3. The common ratio 'r' is the constant value we get when we divide each term by its preceding term. As we found it before, 'r' is 0.1.

Key Concepts

Common Ratio

In the world of geometric series, the "common ratio" is a key player. It's the number by which you multiply each term to get to the next. This constant factor is what makes the series geometric. To find the common ratio, simply divide any term in the series by the term immediately before it. In our exercise example, dividing the second term 0.03 by the first term 0.3 gives us 0.1. If you repeat this process for all subsequent terms, as we've done here, you will obtain the same ratio each time. This confirms that 0.1 is indeed our constant common ratio.

• Formula: Common Ratio = \(r = \frac{a_{n}}{a_{n-1}}\)
• Example: \(r = \frac{0.03}{0.3} = 0.1\)

This concept sustains the essence of a geometric series and allows you to manage the expectation of how terms change throughout the series.

Series Identification

Identifying whether a sequence of numbers is a geometric series requires spotting that elusive common ratio across all terms. A geometric series is identified by a sequence where each term is obtained by multiplying the previous term with a non-zero constant, which is the common ratio. This consistent pattern means that once you find this ratio between two consecutive terms, it holds for every pair of terms in the sequence. For example, if a series starts as 1, 2, 4, 8, each step is multiplied by 2, making 2 the common ratio. In our exercise, by consistently finding the common ratio of 0.1 between terms, we could affirm the series is geometric. This pattern recognition step is crucial for identifying and working with series, particularly within boundaries set by mathematical concepts.

• Recognize pattern through multiplication
• Constant ratio between any consecutive terms
• Essential for further analysis of series properties

First Term of a Series

In any geometric series, knowing the first term—or "the starting point"—is essential. This initial term is typically denoted as "\(a\)" in mathematical terms. It serves as the foundation from which all other terms are derived using the common ratio. Return to our exercise, the first term "\(a\)" is 0.3. From this term, each subsequent term can be calculated by multiplying it repeatedly by the common ratio identified as 0.1. Therefore, it holds immense significance, as it establishes the sequence.

• The initial value in the series
• Used in forming other terms via common ratio
• Integral part of the geometric series formula: \( S_n = a \frac{1-r^n}{1-r} \) where \(S_n\) is the sum of the first \(n\) terms, \(r\) is the common ratio.

Understanding the role and determination of the first term aids significantly in simplifying complex problems involving sequences.