Problem 24 - Precalculus with Limits

Question

In Exercises 17 - 28, write the first five terms of the geometric sequence \( a_1 = 2, r = \pi \)

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are \( 2, 2\pi, 2\pi^2, 2\pi^3, 2\pi^4 \).

Step 1: Determine the first term

The first term is given, \( a_1 = 2 \).

Step 2: Find the second term

To find the second term, we use the formula for a geometric sequence \( a_n = a_1 \cdot r^{(n-1)} \). For the second term where \( n = 2 \), we substitute the given values \( a_2 = 2 \cdot \pi^{(2-1)} = 2\pi \).

Step 3: Find the third term

For the third term where \( n = 3 \), using the formula \( a_3 = 2 \cdot \pi^{(3-1)} = 2 \pi^2 \).

Step 4: Find the fourth term

For the fourth term where \( n = 4 \), using the formula \( a_4 = 2 \cdot \pi^{(4-1)} = 2 \pi^3 \).

Step 5: Find the fifth term

For the fifth term where \( n = 5 \), using the formula \( a_5 = 2 \cdot \pi^{(5-1)} = 2 \pi^4 \).

Key Concepts

Geometric Progression

A geometric progression, or 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 called the common ratio. It is a mathematical concept used to describe a pattern of numbers where each is a product of a constant factor and the term that precedes it. This type of progression can represent a wide variety of phenomena, such as interest compounding in finance or certain aspects of population growth.For example, in the exercise given, we begin with an initial term of 2, and the common ratio is \( \pi \). So, to generate consecutive terms, we multiply the preceding term by \( \pi \), resulting in the sequence: 2, \( 2 \cdot \pi \), \( 2 \cdot \pi^2 \), \( 2 \cdot \pi^3 \), \( 2 \cdot \pi^4 \), and so on. The beauty of a geometric sequence lies in its predictable pattern, useful in various applications, from science to finance.

Sequence of Numbers

A sequence of numbers is an ordered list of numbers following a particular pattern or rule. Sequences appear in numerous areas of mathematics and can be finite or infinite. Within the realm of sequences, a geometric sequence is particularly interesting due to its multiplicative nature.

Identifying Sequences

To identify the type of sequence, you first look at the relationship between consecutive terms. If you notice that you multiply by a certain constant to move from one term to the next, you're dealing with a geometric sequence. In our example, multiplying by \( \pi \) identifies our sequence as geometric.

Writing Sequence Terms

The formality of sequences is expressed in terms of formulas. The general formula for the nth term of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio. This formula provides a direct method to calculate any term in the sequence without listing all preceding terms.

Understanding the structure of number sequences is paramount in subjects like calculus and algebra, where the behavior of functions and series is analyzed deeply.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This concept is often linked with geometric sequences where the common ratio is greater than one. This type of growth is not linear, but rather increases more rapidly as time progresses.The geometric sequence from the exercise can also illustrate exponential growth if we consider each term to represent a time interval. Starting with \( a_1 = 2 \) and a common ratio \( r = \pi \), which is greater than one, implies that each subsequent term will be larger than the previous one, showcasing an exponential increase.Exponential growth is widely observable in nature and technology, such as in populations, infectious diseases, and computing power. Recognizing and understanding this concept can provide insights into trends and patterns, enabling more accurate predictions and better-informed decisions.