Problem 20 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence is geometric, find the common ratio. $$e^{2}, e^{4}, e^{6}, e^{8}, \dots$$

Step-by-Step Solution

Answer

Yes, the sequence is geometric with a common ratio of \( e^{2} \).

Step 1: Determine if the sequence is geometric

A geometric sequence is a sequence where each term after the first is the product of the previous term and a constant called the common ratio \( r \). To check if the sequence \( e^{2}, e^{4}, e^{6}, e^{8} \) is geometric, calculate the ratio between consecutive terms.

Step 2: Calculate the first ratio

Find the ratio between the second term \( e^{4} \) and the first term \( e^{2} \): \[ r_1 = \frac{e^{4}}{e^{2}} = e^{4-2} = e^{2}\]

Step 3: Calculate the second ratio

Find the ratio between the third term \( e^{6} \) and the second term \( e^{4} \):\[ r_2 = \frac{e^{6}}{e^{4}} = e^{6-4} = e^{2}\]

Step 4: Calculate the third ratio

Find the ratio between the fourth term \( e^{8} \) and the third term \( e^{6} \):\[r_3 = \frac{e^{8}}{e^{6}} = e^{8-6} = e^{2}\]

Step 5: Determine if all ratios are equal

All the calculated ratios are equal: \( r_1 = r_2 = r_3 = e^{2} \). Since the ratios between consecutive terms are identical, the sequence is indeed geometric.

Step 6: State the common ratio

The common ratio of the geometric sequence \( e^{2}, e^{4}, e^{6}, e^{8} \) is \( e^{2} \).

Key Concepts

Common Ratio

In the world of geometric sequences, the term "common ratio" plays a crucial role. The common ratio, denoted by \( r \), is the constant factor that you multiply by any term in the sequence to get the next term. It's like the driving force behind a geometric sequence. The sequence will continue to grow or shrink consistently by this same factor.

To understand the concept better, consider the sequence presented in the exercise: \( e^{2}, e^{4}, e^{6}, e^{8} \). To determine if these are terms of a geometric sequence, we calculate the ratio between each pair of consecutive terms. In this case:

• \( r_1 = \frac{e^{4}}{e^{2}} = e^{4-2} = e^{2}\)
• \( r_2 = \frac{e^{6}}{e^{4}} = e^{6-4} = e^{2}\)
• \( r_3 = \frac{e^{8}}{e^{6}} = e^{8-6} = e^{2}\)

All these ratios are equal, confirming that the common ratio is \( e^{2} \). Thus, confirming that this sequence is geometric and showcasing how the concept of a common ratio helps us identify geometric sequences.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers, and understanding them is fundamental in mathematics. A sequence can be arithmetic, geometric, or even neither, depending on its pattern of formation.

A geometric sequence, in particular, involves multiplication by a constant number, known as the common ratio, to progress from one term to the next. In our example sequence, \( e^{2}, e^{4}, e^{6}, e^{8} \), each term is the previous term multiplied by \( e^{2} \), demonstrating the characteristics of a geometric sequence.

Sequences not only help in solving mathematical problems but also find applications in fields like finance (for calculating compound interest), physics (for describing wave patterns), and computer science (for algorithm complexities). Knowing how to identify the kind of sequence you are dealing with is therefore incredibly important.

Exponential Functions

Exponential functions are mathematical functions that can describe real-world processes such as growth or decay. The standard form of an exponential function is \( f(x) = a \cdot b^x \), where \( a \) is the initial amount and \( b \) is the growth factor.

A geometric sequence is closely related to exponential functions because the terms in the sequence can be seen as function values at equally spaced points. For the sequence \( e^{2}, e^{4}, e^{6}, e^{8} \), each term can be represented in the form \( e^{2n} \), where \( n \) represents the position of the term in the sequence. If you think of these as values of a function at these specific points, you're essentially working with an exponential function.

Understanding these relationships can provide deeper insights into how mathematical sequences develop and change over time. Both geometric sequences and exponential functions model situations in which numbers increase or decrease rapidly, often finding real-world applications in areas such as population growth, radioactive decay, and more.