Problem 31 - Algebra and Trigonometry

Question

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(1,-1,1,-1,1,-1, \ldots\)

Step-by-Step Solution

Answer

The nth term is given by \( a_n = (-1)^{n+1} \).

Step 1: Observe the Pattern

Examine the first few terms of the sequence: 1, -1, 1, -1, 1, -1, ... Notice that the terms alternate between 1 and -1.

Step 2: Determine the Cycle Period

Identify the cycle in the pattern. Here, the cycle is 2 terms long: every even-indexed term is -1 and every odd-indexed term is 1.

Step 3: Define the nth term

Express the nth term using a formula that captures the alternation. By the observed pattern, we use \[ a_n = (-1)^{n+1} \]

Step 4: Verify the Formula

Check a few terms to ensure the formula is correct.- For n=1: \( a_1 = (-1)^{1+1} = (-1)^2 = 1 \) - For n=2: \( a_2 = (-1)^{2+1} = (-1)^3 = -1 \) - For n=3: \( a_3 = (-1)^{3+1} = (-1)^4 = 1 \) Thus, the formula holds true for the given sequence.

Key Concepts

alternating sequences

An **alternating sequence** is a sequence of numbers where the terms alternate between different values. In simpler terms, consecutive terms switch back and forth between two values.
In the given exercise, the sequence alternates between 1 and -1.
Understanding alternating sequences is important for spotting patterns in sequences.

• **Observation**: You often start by observing a few terms to understand the repetitive cycle.
• **Pattern Recognition**: Notice if the terms switch between two consistent values.
• **Application**: Use this pattern to predict future terms.

In our example, the sequence is: `1, -1, 1, -1, 1, -1, ...`.
Every odd term is `1`, and every even term is `-1`.
Once we identify this alternating behavior, we can describe the entire sequence succinctly.

pattern identification

Recognizing a pattern is crucial for sequences. It involves investigating the terms to find a repeating cycle or rule.
In our exercise, we identified the alternating pattern: 1, -1, 1, -1, 1, -1, ....
We then realized:

• The sequence has a cycle period of 2 terms.
• Odd-positioned terms are always 1.
• Even-positioned terms are always -1.

**Steps to Identify Patterns**:

• **Examine Initial Terms**: Start with the first few terms (e.g., 1, -1, 1, -1).
• **Look for Repetitions**: Identify cycles or repeated values (e.g., cycles every 2 terms).
• **Determine General Rule**: Establish a rule for any nth term using the identified pattern.

In our case, we formulated the nth term using the exponent pattern \( a_n = (-1)^{n+1} \).
This formula captures the essence of the sequence.

formula verification

Once we have a formula for the nth term, it’s vital to verify it.
Verification ensures that the formula is accurate for all terms.
**Our Formula**: \( a_n = (-1)^{n+1} \).
**Verification Steps**:

• **Plug in Values**: Substitute different n values into the formula to check results.
• **Check Consistency**: Compare the results with the observed sequence terms.

**Example Verification**:
- For n = 1, \( a_1 = (-1)^{1+1} = (-1)^2 = 1 \)
- For n = 2, \( a_2 = (-1)^{2+1} = (-1)^3 = -1 \)
- For n = 3, \( a_3 = (-1)^{3+1} = (-1)^4 = 1 \).
Each example returns the correct sequence term.
These steps confirm our formula is correct and accurately represents the given sequence.
Verification solidifies our understanding and confirms our pattern findings.