Problem 20 - Precalculus with Limits

Question

In Exercises 15 - 22, write the first five terms of the sequence.Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that \( n \) begins with 1. \( a_n = 2^{n - 1} \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 1, 2, 4, 8, 16. The sequence is not arithmetic, so there is no common difference.

Step 1: Find the sequence's first five terms

To find the first five terms of the sequence, substitute the values 1, 2, 3, 4, and 5 for n in the formula \( a_n = 2^{n - 1} \). After solving, the first five terms are 1, 2, 4, 8, 16.

Step 2: Determine if the sequence is arithmetic

To assess if the sequence is arithmetic, find the difference between each consecutive term. The differences are 1 (2-1), 2 (4-2), 4 (8-4), and 8 (16-8). The differences are not constant, hence, the sequence is not arithmetic.

Step 3: Identify the common difference

As the sequence isn't arithmetic, there is no common difference to find. If it were arithmetic, you would take the difference between any two terms, and that would yield the common difference.

Key Concepts

Common Difference

In arithmetic sequences, the term **common difference** refers to the constant number that is added to each term to get the next term in the sequence. This number remains the same throughout the entire series of terms. If you suspect a sequence might be arithmetic, you can check this by subtracting any term from the next one. For instance:

• If the sequence is 2, 4, 6, 8, the common difference is 2.
• Substracting 2 from 4 gives 2, and subtracting 4 from 6 also gives 2, showing a constant difference.

If a sequence has a non-constant difference between its terms, like in the exercise with the sequence 1, 2, 4, 8, 16, then it is not arithmetic. In such cases, there is no common difference. Understanding whether a sequence has a common difference helps in identifying the type of sequence it is.

Terms of a Sequence

A **sequence** is essentially a list of numbers arranged in a specific order. Each number in this list is called a **term**. It's crucial to be able to identify and calculate the terms, especially the initial ones, as these lay the groundwork for understanding the sequence as a whole.
To obtain the first five terms of a sequence like in our exercise, you substitute positive integers starting from 1 into the sequence's formula. In the formula given as \[ a_n = 2^{n - 1} \]if you substitute 1 for \( n \), you get:

• \( a_1 = 2^{1-1} = 2^0 = 1 \)
• \( a_2 = 2^{2-1} = 2^1 = 2 \)
• \( a_3 = 2^{3-1} = 2^2 = 4 \)
• \( a_4 = 2^{4-1} = 2^3 = 8 \)
• \( a_5 = 2^{5-1} = 2^4 = 16 \)

This results in the sequence of 1, 2, 4, 8, 16. Understanding these terms allows you to analyze the entire pattern of the sequence.

Exponential Sequences

An **exponential sequence** grows significantly faster than an arithmetic sequence. In such sequences, each term is a result of multiplying the previous term by a fixed, often large, factor. These sequences follow a pattern where the terms increase or decrease in a geometric fashion.
The sequence from the original exercise, described by \[ a_n = 2^{n - 1} \]is a classic example of an exponential sequence. Each term is derived from raising a constant base (in this case, 2) to the power of an increasing integer (\( n-1 \)). This results in terms that double each time, which is characteristic of exponential growth:

• 1, 2, 4, 8, 16, ...
• From 1 to 2 is a multiplication by 2, from 2 to 4 is again a multiplication by 2, and so on.

Exponential sequences are crucial in many areas, from finance, where they model compound interest, to computer science, especially in algorithms' analysis, highlighting their importance in real-world applications.