Problem 15 - 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. \( a_n = 5 + 3n \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 8, 11, 14, 17, 20. The sequence is arithmetic with a common difference of 3.

Step 1: Compute the first five terms

Use the provided sequence \( a_n = 5 + 3n \) formula to compute the first five terms. All we need to do is substitute the values 1, 2, 3, 4, and 5 for \( n \).

Step 2: Determine if the sequence is arithmetic

Check if the sequence is arithmetic by verifying if the difference between succeeding terms is constant. This is done by subtracting successive terms: \( a_2 - a_1 \), \( a_3 - a_2 \), \( a_4 - a_3 \), and \( a_5 - a_4 \).

Step 3: Determine the common difference

If the sequence is arithmetic, then the difference between any successive terms computed in step 2 should be equal. This constant value is the common difference.

Key Concepts

Arithmetic Sequence Terms

An arithmetic sequence is a list of numbers with a specific pattern: each term after the first is created by adding a constant value, known as the common difference. In the exercise provided, the task is to determine the first five terms of the sequence with the formula a_n = 5 + 3n. Here, it's clear that the sequence starts with a 5 when n=1, and increases by 3 for every additional unit of n. So, for n=2, the term would be 8 (since 5 + 3(2) = 11), and for n=3, the term would be 11 (because 5 + 3(3) = 14), and so on.

Identifying the terms of an arithmetic sequence involves this pattern of addition. You start with the initial term and add the common difference repeatedly to find subsequent terms. If we continue this method, we would find that the first five terms of the exercise's sequence are 8, 11, 14, 17, and 20. These terms are essential for understanding the structure and nature of the sequence.

Common Difference

The common difference is a key concept in an arithmetic sequence. It's the consistent interval or difference between consecutive terms of the sequence. To determine if a sequence is arithmetic, you need to check whether this difference is constant. In the solution steps, calculating (a_2 - a_1), (a_3 - a_2), (a_4 - a_3), and (a_5 - a_4) will reveal the common difference.

From the sequence given in the exercise a_n = 5 + 3n, we subtract each term from the following term and find that the common difference is 3. It's what we add to each term to get to the next one, exemplified by the constant addition of 3 to the previous term. This consistency of subtraction results confirms that we are indeed dealing with an arithmetic sequence. Recognizing the common difference quickly allows for predictions about the sequence and easier calculations of any terms within it.

Sequence Formula

Understanding the formula for an arithmetic sequence is essential for finding any term in the sequence efficiently. The general form of an arithmetic sequence formula is a_n = a_1 + (n - 1)d, where a_n is the nth term of the sequence, a_1 is the first term, n is the term number, and d is the common difference.

In our exercise, the sequence formula given is a_n = 5 + 3n. Let's align this with the general form: You can see that the first term (a_1) is 5, and the common difference (d) is 3. This formula allows direct computation of any term in the sequence without having to build the sequence from the start to the desired term. It's like a shortcut to any point in the numerical pattern, showcasing the beauty of mathematics in describing patterns in a simple, universal language.