Problem 32 - Precalculus: Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\). $$a_{n}=3+(-1)^{n} n$$

Step-by-Step Solution

Answer

The first five terms are 2, 5, 0, 7, and -2; the sequence is not arithmetic.

Step 1: Understand the Sequence Formula

The given formula for the sequence is \(a_{n} = 3 + (-1)^{n} n\). This means for each term \(a_{n}\), whether \((-1)^{n}\) is positive or negative depends on whether \(n\) is odd or even, respectively.

Step 2: Calculate the First Five Terms

To find the first five terms, substitute \(n = 1, 2, 3, 4, 5\) into the sequence formula:- When \(n=1\), \(a_1 = 3 + (-1)^{1} \times 1 = 3 + (-1) = 2\).- When \(n=2\), \(a_2 = 3 + (-1)^{2} \times 2 = 3 + 2 = 5\).- When \(n=3\), \(a_3 = 3 + (-1)^{3} \times 3 = 3 - 3 = 0\).- When \(n=4\), \(a_4 = 3 + (-1)^{4} \times 4 = 3 + 4 = 7\).- When \(n=5\), \(a_5 = 3 + (-1)^{5} \times 5 = 3 - 5 = -2\).Thus, the first five terms are 2, 5, 0, 7, and -2.

Step 3: Determine if the Sequence is Arithmetic

An arithmetic sequence has a constant difference \(d\) between consecutive terms. Calculate the differences:- \((a_2 - a_1) = 5 - 2 = 3\)- \((a_3 - a_2) = 0 - 5 = -5\)- \((a_4 - a_3) = 7 - 0 = 7\)- \((a_5 - a_4) = -2 - 7 = -9\)The differences are not constant, so the sequence is not arithmetic.

Key Concepts

Sequence Formula

A sequence formula is an important tool for understanding how to generate the terms of a sequence. In this exercise, the given sequence formula is \(a_{n} = 3 + (-1)^{n} n\). Let's break it down so it's easier to grasp. The expression \((-1)^{n}\) plays a critical role because it determines whether each term has an added or subtracted value. When \(n\) is an odd number, \((-1)^{n}\) equals \(-1\), and when \(n\) is even, \((-1)^{n}\) is \(+1\).

This alternating pattern suggests that the sequence won't have a uniform progression, which is a key aspect of this specific sequence formula. Understanding how to apply the sequence formula is vital for finding the terms, as it allows you to substitute any positive integer \(n\) to calculate the corresponding term \(a_n\). So, you can deduce the specific values for each term as we've done in the calculation of the first five terms: 2, 5, 0, 7, and -2.

Common Difference

The concept of a common difference is crucial for identifying arithmetic sequences. An arithmetic sequence is defined by having a constant, or common, difference between all consecutive terms. In simpler terms, the difference between every pair of terms (the consecutive ones) is consistent. For example, in the sequence 1, 3, 5, 7, the common difference is 2 because each term increases by 2.

In our given sequence, after calculating the first five terms as 2, 5, 0, 7, and -2, we checked for a common difference by subtracting each term from the subsequent one:

• \(5 - 2 = 3\)
• \(0 - 5 = -5\)
• \(7 - 0 = 7\)
• \(-2 - 7 = -9\)

These differences vary, meaning there isn't a consistent difference. Hence, this sequence is not arithmetic because it doesn't maintain a steady progression. Understanding this process is critical when learning about sequences, as it helps in classifying the sequences correctly.

Nth Term

The \(n\)th term of a sequence allows us to generalize and calculate any term within the sequence. Usually, for arithmetic sequences, the \(n\)th term is expressed in the form \(a_{n} = a + (n-1) \, d\), where \(a\) is the first term and \(d\) is the common difference. This formula is fundamental in specifying a particular pattern that arithmetic sequences follow.

However, the given sequence in our problem doesn't follow a traditional arithmetic pattern. Its formula, \(a_{n} = 3 + (-1)^{n} n\), incorporates both positive and negative changes in value due to the \((-1)^{n}\) factor, making it non-arithmetic. Therefore, while the concept of \(a_{n}\) and the standard form of arithmetic sequences might not apply here, understanding the typical structure aids in discerning how this sequence operates differently.

By appreciating the role of an \(n\)th term, one can better grasp how sequences are formed and calculated across different types.