Problem 53 - Algebra and Trigonometry

Question

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=2^{n} $$

Step-by-Step Solution

Answer

The given sequence \(a_{n}=2^{n}\) is a geometric sequence with a common ratio of 2.

Step 1: Analyze the given sequence

By direct observation, it's evident that the term increases by multiplication, not by addition of a constant number, therefore, it does not represent an arithmetic sequence. The provided sequence is \( a_{n} = 2^{n} \). The multiplication factor between the terms is the base of the power, suggesting that this might be a geometric sequence.

Step 2: Confirm if it is a Geometric sequence

A geometric sequence has each term being the previous term multiplied by a fixed number, called the common ratio. We will pick two consecutive terms of our sequence and try dividing the later term by the earlier one to check if the ratio remains constant. Let's pick the first and second term for simplicity, where \( n = 1 \) and \( n = 2 \). Hence, \( a_{2}/a_{1} = 2^{2}/2^{1} = 2\). The ratio is constant and equal to 2, confirming that our sequence is geometric.

Step 3: Identify the common ratio of the geometric sequence

We found that the common ratio of this sequence is 2, as for each step from \(a_n\) to \(a_{n+1}\), we multiply by 2. This is because \(2^{n+1}/2^{n} = 2\). Therefore, the common ratio is 2.

Key Concepts

Common Ratio

The common ratio is an essential element in the study of geometric sequences. In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. This can be expressed as:

• First term: \(a_1\)
• Second term: \(a_2 = a_1 imes r\)
• Third term: \(a_3 = a_2 imes r = a_1 imes r^2\)

The formula shows that each new term is generated by continuously multiplying the previous term by this common ratio \(r\).
In the sequence example \(a_n = 2^n\), we detect a consistent multiplication factor when exploring consecutive terms. To determine the common ratio: take two consecutive terms such as \(a_2\) and \(a_1\) then divide the later by the former:\[\frac{a_2}{a_1} = \frac{2^2}{2^1} = 2\]Thus, the common ratio \(r\) is 2, confirming the sequence as geometric because the ratio stays constant throughout.

Arithmetic Sequence

An arithmetic sequence is a different type of number sequence where each term is generated by adding a fixed, constant number to the previous term. This constant is known as the common difference. The structure of an arithmetic sequence is:

• First term: \(a_1\)
• Second term: \(a_2 = a_1 + d\)
• Third term: \(a_3 = a_2 + d = a_1 + 2d\)

Here, \(d\) represents the common difference across the sequence.
In this lesson, the sequence \(a_n = 2^n\) is not considered arithmetic as it grows by multiplication instead of simple addition. No constant is added to produce subsequent terms, making it impossible to establish a common difference.

Sequence Analysis

Sequence analysis involves determining the nature of a sequence—whether it is arithmetic, geometric, or neither. This process begins by understanding the rules defining the sequence. Sequence analysis helps us:

• Identify patterns in the sequence of numbers.
• Determine whether a sequence follows a specific rule.
• Calculate terms beyond those initially provided, using the identified pattern.

In this example, the process began by observing the general term \(a_n = 2^n\). Detecting that the terms \(2^{1}, 2^{2}, 2^{3},...\) are formed by multiplying the preceding term by 2, hints towards a geometric pattern. Through careful computation, it was confirmed that the sequence is geometric, and has a consistent common ratio of 2. This form of analysis is critical not only to identify the type of sequence but also to enhance understanding of mathematical relationships and growth patterns present in everyday applications.