Problem 9 - 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 " \(r\) ". $$ \frac{3}{2}-\frac{3}{4}+\frac{3}{8}-\cdots-\frac{3}{2^{6}} $$

Step-by-Step Solution

Answer

The sum is a geometric series. The first term 'a' is \( \frac{3}{2} \) and the common ratio 'r' is \(-\frac{1}{2} \).

Step 1: Identify the Pattern

Look at the series and try to identify any patterns. The series given is \( \frac{3}{2}, -\frac{3}{4}, \frac{3}{8}, \ldots \). The sign is alternating (-,+) which enforces the notion that this could probably be a geometric series.

Step 2: Calculate the Ratio

Next, verify if the series is really geometric by checking if the ratio between consecutive terms is constant. Calculate the ratio like this: \( \frac{-\frac{3}{4}}{\frac{3}{2}} = -\frac{1}{2} \), \( \frac{\frac{3}{8}}{-\frac{3}{4}} = -\frac{1}{2} \) and so on.

Step 3: Declare a and r

Since the ratio between consecutive terms is constant, we confirm that this is indeed a geometric series. Therefore, \(a\) represents the first term of the series, which is \( \frac{3}{2} \) and the common ratio \( r \) is \(-\frac{1}{2} \).

Key Concepts

Common Ratio

In a geometric series, the common ratio is a vital component. It determines the relationship between terms in the sequence. To establish whether a series is geometric, calculate the ratio between two consecutive terms. If this ratio stays constant throughout the series, then it is indeed a geometric series. In our example, we calculate the common ratio to be \( r = -\frac{1}{2} \), by dividing each term by the term before it. Hence, \(-\frac{3}{4} \div \frac{3}{2} = -\frac{1}{2}\), and similarly, \(\frac{3}{8} \div (-\frac{3}{4}) = -\frac{1}{2}\). The common ratio answering this problem is uniform, confirming that the sequence is geometric.

Alternating Series

An alternating series is one in which the signs of the terms alternate between positive and negative. Observing the series \( \frac{3}{2}, -\frac{3}{4}, \frac{3}{8}, \ldots \), it is evident that each term switches its sign compared to the previous one. This characteristic is due to the negative common ratio \( -\frac{1}{2} \). Here, the negative ratio ensures that each term flips in sign, producing this distinctive back-and-forth in sequence. Understanding the signs can help anticipate what follows in the sequence, crucial for identifying geometric properties.

Pattern Identification

Recognizing patterns in sequences can guide you to discern the type of series you're working with. Begin by observing if there's a consistent method or trait like alternation in signs or a uniform multiplier between terms. In the exercise, despite the alternating signs, there's a pattern in the numerators and denominators which further suggests the relationship. The pattern can be identified visually before doing extensive calculations, setting the stage for understanding that the series is geometric.

Calculating Ratios

Once you've identified a potential pattern, verify it by calculating the ratios. This calculation is essential to confirm a geometric sequence. Divide the second term by the first term, the third by the second, and continue with subsequent pairs. This operation should yield the same value consistently to affirm a geometric series. For instance, in our series, using successive terms, the ratio consistently calculated as \(-\frac{1}{2}\) verified our initial assumption that the series is geometric.