Problem 17 - 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. \( a_n = 3 - 4\left(n - 2\right) \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 7, 3, -1, -5, -9. The sequence is arithmetic and the common difference is -4.

Step 1: Calculating the First Terms

Firstly, we need to find out the first five terms of the sequence. To do this, we simply plug in the values \( n = 1, 2, 3, 4, 5 \) into the function and calculate the results. \nFor \( n = 1 \): \n\( a_1 = 3 - 4\left(1 - 2\right) = 3 - 4*(-1) = 7\n \)For \( n = 2 \): \n\( a_2 = 3 - 4\left(2 - 2\right) = 3\n \)For \( n = 3 \): \n\( a_3 = 3 - 4\left(3 - 2\right) = 3 - 4*1 = -1\n \)For \( n = 4 \): \n\( a_4 = 3 - 4\left(4 - 2\right) = 3 - 4*2 = -5\n \)For \( n = 5 \): \n\( a_5 = 3 - 4\left(5 - 2\right) = 3 - 4*3 = -9\n \)

Step 2: Identifying the Sequence Type

Now that we have the first five terms, we need to find out if the sequence is arithmetic. An arithmetic sequence is defined as a sequence of numbers where the difference between any two successive members is a constant. In this case, by subtracting each term from the succeeding one, we get:\( 7 - 3 = 4 \), \( 3 - (-1) = 4 \), \( -1 - (-5) = 4 \), \( -5 - (-9) = 4 \). \nThe differences between the terms are constant and equal to 4, so the sequence is arithmetic.

Step 3: Finding the Common Difference

The common difference of an arithmetic sequence is the constant difference between successive terms. We already determined that, in this case, the common difference is \( -4 \), that is, each term subtracts 4 from the previous one.

Key Concepts

Sequence of Numbers

When we talk about a sequence of numbers, we're referring to an ordered list of numbers that follow a specific pattern or rule. It's like a set of numbers that are in a particular order for a reason, not just randomly thrown together. For instance, the sequence 2, 4, 6, 8, 10 is formed by adding 2 to the starting number each time.

Looking at our example exercise, the sequence is determined by the formula: a_n = 3 - 4(n - 2). To identify the members of this sequence, we start at n = 1 and increase n by 1 each time. This is essential for understanding how sequences are formed and how to calculate their terms.

Common Difference

In the world of sequences, the common difference is a key concept that sets arithmetic sequences apart from others. It is the consistent interval or gap between consecutive numbers in an arithmetic sequence. Think of it as the 'step size' that you take from one term to the next. If you're adding or subtracting the same value each time you move to the next number, then you have a common difference.

From the provided solution, the common difference is found by subtracting one term from the next, and if this difference is the same throughout, the sequence is arithmetic. In our exercise, 7 - 3 = 4, 3 - (-1) = 4, and so on, shows a consistent common difference of -4. Identifying the common difference is crucial for understanding the pattern within an arithmetic sequence and for making predictions about other terms within the sequence.

Term of a Sequence

The term of a sequence refers to an individual member of the sequence. When we talk about the 'first term', 'second term', and so on, we're pointing out specific positions in the sequence. Just like the chapters of a book or steps on a ladder, each term occupies a unique spot in the lineup defined by the sequence's rule.

In arithmetic sequences like the one in our exercise, each term is generated by applying a specific formula. With a_n = 3 - 4(n - 2), for each n (which represents the term's position), we calculate the value of the term. It's important to understand both the position of a term in a sequence and how to compute its value because this is the foundation for working with any type of numerical sequence.