Problem 26 - Algebra and Trigonometry

Question

21-26 \(\approx\) 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 sequence is not arithmetic; first five terms: 2, 5, 0, 7, -2.

Step 1: Identify pattern of sequence

Given the sequence formula is \( a_n = 3 + (-1)^n n \). To find the first five terms, we will substitute \( n = 1, 2, 3, 4, 5 \) into this formula.

Step 2: Calculate the first term

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

Step 3: Calculate the second term

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

Step 4: Calculate the third term

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

Step 5: Calculate the fourth term

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

Step 6: Calculate the fifth term

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

Step 7: Check if the sequence is arithmetic

The first five terms of the sequence are 2, 5, 0, 7, -2. The differences are calculated as follows: Positive difference from 2 to 5: 5 - 2 = 3 Decrease from 5 to 0: 0 - 5 = -5 Increase from 0 to 7: 7 - 0 = 7 Decrease from 7 to -2: -2 - 7 = -9 Since these differences are not constant, the sequence is not arithmetic.

Key Concepts

Sequence Patterns

A sequence is an ordered list of numbers that follow a particular pattern. Understanding patterns in sequences helps predict future terms and understand the structure of the sequence. In sequences, patterns can be either straightforward or complex. A straightforward sequence is one where you repeatedly apply the same operation to generate the next term. For example, adding the same number each time. Complex patterns, like the one in the formula \( a_n = 3 + (-1)^n n \), alternate the operations, creating a sequence that may not seem immediately predictable. In our example, the pattern uses an alternating process, such that:

• When \( n \) is odd, the sequence subtracts the value \( n \).
• When \( n \) is even, the sequence adds the value \( n \).

This alternating pattern makes it essential to calculate each term individually, as the pattern does not follow a simple arithmetic progression.

Common Difference

The concept of a common difference is central to determining if a sequence is arithmetic. The common difference is the value added to each term to reach the next one. When this difference is the same between every consecutive term, the sequence is termed 'arithmetic.'In our exercise, to determine if the sequence \( a_n = 3 + (-1)^n n \) is arithmetic, we calculated differences between consecutive terms:

• First to second term: \( 5 - 2 = 3 \)
• Second to third term: \( 0 - 5 = -5 \)
• Third to fourth term: \( 7 - 0 = 7 \)
• Fourth to fifth term: \( -2 - 7 = -9 \)

These differences are not constant, so the sequence does not have a common difference.

nth Term Formula

The nth term formula calculates any term in a sequence without listing all previous terms. This formula is vital for quickly finding specific terms in large sequences. An arithmetic sequence uses the nth term formula \( a_n = a_1 + (n-1) \cdot d \), where:

• \( a_n \) is the nth term.
• \( a_1 \) is the first term.
• \( d \) is the common difference.

For a sequence to apply this formula, it must be arithmetic, meaning the difference between terms stays constant.In our example, the sequence \( a_n = 3 + (-1)^n n \) does not fit this criteria since it does not have a common difference. Therefore, it does not utilize the arithmetic sequence's nth term formula. This example highlights the importance of understanding a sequence's pattern before applying specific formulas.