Problem 22 - Precalculus: Mathematical for Calculus

Question

Find the first five terms of the sequence and determine if 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 sequence is not arithmetic, as its differences are inconsistent: 3, -5, 7, -9.

Step 1: Calculate the First Term

Substitute \( n = 1 \) into the formula to find the first term: \[ a_1 = 3 + (-1)^1 \times 1 = 3 - 1 = 2. \] So, the first term is 2.

Step 2: Calculate the Second Term

Substitute \( n = 2 \) into the formula to find the second term: \[ a_2 = 3 + (-1)^2 \times 2 = 3 + 2 = 5. \] So, the second term is 5.

Step 3: Calculate the Third Term

Substitute \( n = 3 \) into the formula to find the third term: \[ a_3 = 3 + (-1)^3 \times 3 = 3 - 3 = 0. \] So, the third term is 0.

Step 4: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula to find the fourth term: \[ a_4 = 3 + (-1)^4 \times 4 = 3 + 4 = 7. \] So, the fourth term is 7.

Step 5: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula to find the fifth term: \[ a_5 = 3 + (-1)^5 \times 5 = 3 - 5 = -2. \] So, the fifth term is -2.

Step 6: Identify the Pattern and Determine Arithmetic Nature

The first five terms are 2, 5, 0, 7, -2. Compare differences: 5-2=3, 0-5=-5, 7-0=7, -2-7=-9. Since the differences are not consistent, this sequence is not arithmetic.

Key Concepts

Sequence Analysis

Understanding and identifying sequences is crucial. A sequence is essentially just an ordered list of numbers. In this exercise, we are given the formula for the nth term of a sequence: \[a_{n}=3+(-1)^{n}n\]. Given a formula, we can calculate particular values of the sequence by substituting values in for \( n \). The goal here is to find the first five terms and determine if the sequence follows a certain pattern, like being arithmetic. Each sequence can have its own unique pattern.

• If the pattern involves a constant difference between consecutive terms, it could be an arithmetic sequence.

Sequence analysis starts with observing terms and identifying patterns that define the entire list.

First Five Terms

To fully understand a sequence, especially when provided as a formula, it's helpful to calculate its first few terms. For the given sequence, the process involves substituting in successive integer values for \( n \) and simplifying:

• First Term: \( a_1 = 3 + (-1)^1 \times 1 = 2 \).
• Second Term: \( a_2 = 3 + (-1)^2 \times 2 = 5 \).
• Third Term: \( a_3 = 3 + (-1)^3 \times 3 = 0 \).
• Fourth Term: \( a_4 = 3 + (-1)^4 \times 4 = 7 \).
• Fifth Term: \( a_5 = 3 + (-1)^5 \times 5 = -2 \).

The first five terms of this sequence are 2, 5, 0, 7, and -2. With these terms, you can start looking for patterns or regularities that characterize the sequence.

Common Difference

One main feature of an arithmetic sequence is the common difference. This is the consistent interval between consecutive terms. For our sequence, we examined the consecutive terms:

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

The differences are 3, -5, 7, and -9, clearly not all the same. Therefore, this sequence lacks the required consistency to be classified as arithmetic. In arithmetic sequences, the common difference is always constant, which allows for straightforward pattern prediction.

Nth Term Formula

Understanding how the nth term formula connects to the sequence is vital. Here, the formula given was \( a_{n}=3+(-1)^{n}n \). It combines constant, variable, and alternating components:

• The constant \(3\) provides a baseline value.
• The term \((-1)^n\) introduces alternating signs based on whether \(n\) is even or odd.
• The variable \(n\) itself varies as you progress through the sequence.

Together, they create terms that fluctuate due to the alternating sign impact. This formula and the resulting pattern show how not all sequences will be arithmetic. Sequencing rules in formulas can define unique behaviors, making each sequence development different from others.