Problem 15 - 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 " \(r\) ". $$ 0.2+0.06+0.018+0.0054+0.00162 $$

Step-by-Step Solution

Answer

The sum is a geometric sequence. The first term (a) is 0.2 and the common ratio (r) is 0.3.

Step 1: Identify a possible common ratio

To identify if the sequence is geometric, try to find a common ratio \(r\). This can be done by dividing the second term by the first term, which gives you 0.06 / 0.2 = 0.3

Step 2: Check if the ratio holds

To make sure it is a geometric sequence, also try to divide the subsequent terms and see if they also give the ratio 'r'. 0.018 / 0.06 = 0.3 and 0.0054 / 0.018 = 0.3, 0.00162 / 0.0054 = 0.3. Therefore, the common ratio is indeed constant and equals \(r = 0.3\). Therefore, the sequence is Geometric.

Step 3: Identify 'a' and 'r'

In a geometric sequence, the first term is 'a' and the common ratio is 'r', which has already been identified. Here the first term \(a = 0.2\) and the common ratio \(r = 0.3\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key feature that distinguishes it from other types of sequences. This ratio, denoted by the symbol \(r\), is the factor by which each term in the sequence is multiplied to get the subsequent term.
For instance, if you have a sequence where the terms are \(0.2,\ 0.06,\ 0.018,\ldots\), you find \(r\) by dividing one term by the previous one—like \(0.06 \div 0.2 = 0.3\).

• The value of \(r\) must be constant throughout the sequence.
• If \(r\) changes, then the sequence is not geometric.

In our example, the sequence maintains a consistent common ratio of \(0.3\), confirming its geometric nature.

First Term

The first term in a geometric sequence, denoted as \(a\), is where the sequence starts. It's crucial in defining the sequence's progression. The first term can change the entire sequence even for the same common ratio.
In our specific example, the first term is \(0.2\).

• The entire geometric sequence is constructed based on this initial value.
• Each subsequent term is calculated by multiplying the first term with increasing powers of the common ratio.

Understanding the first term helps set the framework for both calculation and comprehension of the sequence.

Series and Sequences

In mathematics, a series is the sum of the terms of a sequence. When we say we have a geometric sequence, we talk about the ordered list of numbers, but a geometric series is the result of adding these numbers together.

• A sequence, like our example \(0.2,\ 0.06,\ 0.018,\ldots\), can be finite or infinite.
• A series is often used when looking to find the sum of terms in this sequence.

Determining whether a sequence has formed a series answers broader questions about convergence and the possibility of calculating a sum for infinite terms. In the original exercise, recognizing the pattern's continuation helped affirm the geometric sequence's formation, which in turn implies potential calculations for a geometric series from these terms.