Problem 18 - 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. \( a_n = 1 + \left(n - 14\right)4 \)

Step-by-Step Solution

Answer

The first five terms of the sequence are -52, -48, -44, -40, -36. The sequence is arithmetic and the common difference is 4.

Step 1: Find the first five terms of the sequence

Substitute \( n \) = 1, 2, 3, 4, 5 into the formula \( a_n = 1 + \left(n - 14\right)4 \) respectively to get: \( a_1 = 1 + (1 - 14) 4 = -52 \) \( a_2 = 1 + (2 - 14) 4 = -48 \) \( a_3 = 1 + (3 - 14) 4 = -44 \) \( a_4 = 1 + (4 - 14) 4 = -40 \) \( a_5 = 1 + (5 - 14) 4 = -36 \)

Step 2: Determine whether the sequence is arithmetic

A sequence is arithmetic if the differences between the terms are constant. To find this, subtract each term from the one that follows it: \( a_2 - a_1 = -48 - ( -52 ) = 4 \) \( a_3 - a_2 = -44 - ( -48 ) = 4 \) \( a_4 - a_3 = -40 - ( -44 ) = 4 \) \( a_5 - a_4 = -36 - ( -40 ) = 4 \) The differences are constant, so the sequence is arithmetic.

Step 3: Find the common difference

The common difference in an arithmetic sequence is the constant difference between consecutive terms. Since the differences are all 4, the common difference is 4.

Key Concepts

Common Difference

When we talk about arithmetic sequences, one critical component to understand is the common difference. It's the steady interval that separates each term in an arithmetic sequence. This aspect makes the sequence predictable and allows us to easily identify any term in the sequence without listing all the previous ones.

For example, consider the sequence produced by the formula a_n = 1 + (n - 14)4. If we plug in consecutive values of n, we notice that each term is 4 units apart from the next. This uniform spacing is precisely what we mean by a common difference. Calculating it is simple: take any term in the sequence and subtract the one before it. Here, for any n, a_{n+1} - a_n = 4, demonstrating a common difference of 4. This is a hallmark of an arithmetic sequence.

Sequence Terms Calculation

Calculating the terms of a mathematical sequence, particularly an arithmetic sequence, involves using its explicit formula. For an arithmetic sequence, the formula usually has the structure a_n = a_1 + (n - 1)d, where a_1 is the first term, n is the term number, and d is the common difference.

Let's delve into how we determined the first five terms of the provided sequence a_n = 1 + (n - 14)4. By substituting values of n starting from 1, we obtained terms from a_1 to a_5. This process exemplifies a practical application of the sequence's explicit formula to calculate specific terms. By continuing this pattern, we could compute further terms as needed.

Mathematical Sequences

Sequences are fundamental in mathematics, representing ordered lists of numbers. Each number in a sequence is called a 'term.' Among the variety of sequences, one prominent type is the arithmetic sequence. In these sequences, each term after the first is found by adding a constant value, called the common difference, to the previous term.

An arithmetic sequence provides a clear illustration of linear patterns in numerical contexts and is an essential concept in understanding how to navigate series and sums in algebra. The ability to identify and work with these sequences is a valuable skill in various mathematical applications, from solving algebraic problems to understanding real-world phenomena that exhibit linear growth or decay.