Problem 17 - 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 " \(r\) ". $$ p+p^{3}+p^{5}+p^{7}+p^{9} $$

Step-by-Step Solution

Answer

The sum \(p+p^{3}+p^{5}+p^{7}+p^{9}\) is a geometric series with the first term 'a' being \(p\) and the common ratio 'r' being \(p^{2}\).

Step 1: Arranging the Sum

First arrange the sum that matches a potential geometric series. For the sum given: \(p+p^{3}+p^{5}+p^{7}+p^{9}\), we can rewrite it as \(p + p^3 + p^5 + p^7 + p^9 = p(1 + p^{2} + p^{4}+ p^{6} + p^{8})\), as this looks more like a typical geometric series with specifying weights.

Step 2: Identifying whether it is a Geometric Series

Next, determine if it is a geometric series. A geometric series has the condition that each term is the previous term multiplied by a constant. Here the terms inside the bracket are: \(1, p^{2}, p^{4}, p^{6}, p^{8}\). If this series is geometric, the ratio between each successive term should be constant. Dividing each term by the previous one, the ratios are \(p^{2}, p^{2}, p^{2}, p^{2}\) respectively. This indicates that the sum inside the bracket is a geometric series.

Step 3: Identifying the first term 'a' and the common ratio 'r'

Now, determine the first term \(a\) and the common ratio \(r\) of the series. Given the general form of a geometric series is \(a + ar + ar^2 + ar^3\cdots\), we can see here, 'a' is 1 and the common ratio 'r' is \(p^{2}\). These are the values for 'a' and 'r' of the sum inside the bracket. As for the entire sum, the first term 'a' is \(p\) and the common ratio is \(p^{2}\).

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number. This fixed number is known as the common ratio. Imagine you have a set of numbers where each one is generated from the one before it by a simple multiplication. This kind of sequence can grow or shrink very rapidly depending on whether the common ratio is greater or less than one. Here’s how it works:

• Start with the first term, known as 'a'.
• Multiply 'a' by the common ratio 'r' to get the next term.
• Continue multiplying each new term by 'r' to find the subsequent terms.

Geometric sequences are found everywhere in nature and science, such as in the growth of populations or radioactive decay.

Common Ratio

The common ratio in a geometric sequence is the key factor that defines its pattern. It’s the number you multiply by to get from one term to the next. In our exercise, every term inside the bracket after the first was achieved by multiplying by the common ratio, which in this case is \(p^2\).

• A common ratio greater than 1 will make your sequence grow.
• When the common ratio is between 0 and 1, the sequence shrinks.
• If the common ratio is negative, the sequence will alternate in sign.

Understanding the common ratio enables you to predict the future terms of the sequence effectively.

First Term

The first term of a sequence is like the starting point of a journey. In a geometric sequence, it is denoted by 'a'. This term is crucial since every subsequent term is generated from here using the common ratio. For example, in our solution, the first term of the sequence was determined to be 1 for the terms inside the bracket, making each future calculation relative to this initial term.
Similarly, looking at the entire sequence, the first term 'a' is \(p\).

• This single term provides the base for constructing the entire sequence.
• Often, changing the first term will affect the whole sequence pattern.

Knowing the first term is essential for defining and continuing a sequence.

Patterns in Series

Series often exhibit particular patterns that help us understand the structure and predictability of sequences. A geometric series, like the one identified in the exercise, clearly exhibits such regularity by consistently following a set multiplication pattern defined by the common ratio.

• In our case, adding each new term involves a consistent multiplication by \(p^2\).
• The pattern repetition allows for easier computation of the sum of the series.
• By analyzing these patterns, you can extrapolate beyond the visible series and predict subsequent terms.

Recognizing patterns in series is not only crucial for solving mathematical problems but also offers insight into various natural and scientific phenomena.