Problem 54 - College Algebra

Question

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. $$a_{n}=\left(\frac{1}{2}\right)^{n}$$

Step-by-Step Solution

Answer

The sequence \(a_{n}=\left(\frac{1}{2}\right)^{n}\) is geometric with a common ratio of 1/2.

Step 1: Identify the sequence

The sequence given in the exercise is \(a_{n}=\left(\frac{1}{2}\right)^{n}\). This sequence represents the n-th term of the sequence where the first term \(n=1\) is 1/2, the second term \(n=2\) is 1/4, and so on.

Step 2: Check for common difference or ratio

To find out if the sequence is arithmetic or geometric, one needs to check for a common difference or common ratio. When we calculate the common ratio between terms (i.e., divide each term by the previous one), the ratio comes out to be 1/2 which stays constant. For example, the ratio from \(n=2\) to \(n=1\) is \(\left(\frac{1}{4}\right)/\left(\frac{1}{2}\right) = \frac{1}{2}\) and the ratio from \(n=3\) to \(n=2\) is \(\left(\frac{1}{8}\right)/\left(\frac{1}{4}\right) = \frac{1}{2}\). Thus, the common ratio is 1/2.

Step 3: Determine the type of sequence

Once we observe that the common ratio is constant, we determine that this is a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio, which in this case is 1/2.

Key Concepts

Common Ratio

Understanding the common ratio is pivotal when studying geometric sequences. It's the factor by which we multiply each term to get to the next one in a sequence. If you look at a clear-cut example such as the sequence provided, \(a_n=\left(\frac{1}{2}\right)^n\), we can see this concept in action. Here, each term is created by multiplying the previous term by \(\frac{1}{2}\).

To check the common ratio, divide a term in the sequence by its preceding term. If this ratio is consistent between any consecutive terms, then the sequence is geometric. For instance, \(\frac{a_2}{a_1} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\) and \(\frac{a_3}{a_2} = \frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}\). Since this ratio remains unchanged, it confirms that \(\frac{1}{2}\) is indeed the common ratio for the given sequence. Remembering that the consistency of this ratio is the definitive test for a geometric sequence will help you recognize and construct these series with confidence.

Arithmetic Sequence

In contrast to geometric sequences, an arithmetic sequence progresses by adding a constant value to each term to get the next one. This constant value is known as the common difference. For example, in an arithmetic sequence like 3, 6, 9, 12,..., the common difference is 3.

To determine if a sequence is arithmetic, you would subtract a term from the one that follows it. If the result is a constant, that's your common difference. Suppose we have \(b_n\) defined as the n-th term of an arithmetic sequence, then \(b_{n+1} - b_n\) would always equal the same value, the common difference. It's key to recognize that while geometric sequences involve multiplication or division by a common ratio, arithmetic sequences focus on addition or subtraction by a common difference. Understanding this essential distinction will greatly aid in identifying and working with the two types of sequences.

Sequence and Series

Once you're familiar with individual sequences, diving into the broader concept of sequence and series is the logical next step. A sequence is a set of numbers arranged in a specific order following a certain rule, like the geometric or arithmetic sequences we've discussed. A series, on the other hand, is the sum of the terms of a sequence.

For example, if \(a_n\) is a geometric sequence, then the sum \(S_n = a_1 + a_2 + a_3 + ... + a_n\) is a geometric series. Similarly, if \(b_n\) is an arithmetic sequence, \(S_n = b_1 + b_2 + b_3 + ... + b_n\) is an arithmetic series. Understanding series is beneficial for various applications, including calculating interest in finance or analyzing patterns in data sets.

To sum up, the difference between a sequence and a series is that a sequence lists numbers, while a series adds them up. Recognizing whether a series is arithmetic or geometric can significantly simplify the process of finding its sum, especially as both types have formulas to calculate their sums efficiently, leveraging the common difference or common ratio.