Question

Find a formula for the general term \(a_{n}\) of each of the following sequences. $$ \\{1,0,-1,0,1,0,-1,0, \ldots\\} \text { (Hint: Find where } \sin x \text { takes these values) } $$

Short answer

The general term formula is \( a_n = \sin\left(\frac{\pi}{2} + n\pi\right) \).

Step-by-step solution

Identify the Pattern in the Sequence

Examine the given sequence \( \{1, 0, -1, 0, 1, 0, -1, 0, \ldots \} \). Notice that the sequence repeats every four terms with the pattern \( \{1, 0, -1, 0\} \). This suggests periodic behavior.

Relate Sequence to Sine Function

Recognize from the hint that the sine function \( \sin(x) \) achieves the values 1, 0, -1, 0 periodically. Specifically, \( \sin\left(\frac{\pi}{2}\right) = 1, \sin(\pi) = 0, \sin\left(\frac{3\pi}{2}\right) = -1, \sin(2\pi) = 0 \).

Find the Corresponding Argument for Each Term

Using the observations from Step 2, match the sine values to sequence terms. Notice that the sequence terms are given by substitutions of \( x = \frac{\pi}{2} + n\pi \) in the sine function, where \( n = 0, 1, 2, \ldots \).

Derive the General Formula

The formula for the general term \( a_n \) can be expressed using the function: \( a_n = \sin\left(\frac{\pi}{2} + n\pi\right) \). This formula captures the periodic nature of the sequence since \( \sin\left(\frac{\pi}{2} + n\pi\right) \) repeats every four terms as \( \{1, 0, -1, 0\} \).

Key concepts

Periodic Sequences

A periodic sequence is one where the terms repeat after a certain number of steps. In the given sequence \( \{1, 0, -1, 0, 1, 0, -1, 0, \ldots \} \), the pattern repeats every four terms: \( \{1, 0, -1, 0\} \). This repeating cycle is known as the period of the sequence.

Understanding periodic sequences helps us predict future elements without listing all previous terms. In mathematics, being able to identify the period can save a lot of effort, especially in complex problems.

Some characteristics of periodic sequences include:

• A fixed set of values that repeat over and over.
• The ability to wrap the sequence into multiple cycles involving the same pattern.
• Convenient for problems involving symmetry and cyclic behaviors.

By recognizing the periodic nature, you can derive general formulas that simplify the sequence into a more manageable form.

Sine Function in Sequences

The sine function, denoted as \( \sin(x) \), is fundamental in trigonometry. It exhibits a periodic nature similar to sequences like the one we're analyzing.

For the sequence \( \{1, 0, -1, 0\} \), we see the values mirror those of the sine function at specific angles:

• \( \sin\left(\frac{\pi}{2}\right) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
• \( \sin(2\pi) = 0 \)

By aligning these angles with our sequence, we identify the cycle is a sine wave stretched over points where \( x = \frac{\pi}{2} + n\pi \).

The power of correlating the sine function to sequences is in its predictable periodicity. It's valuable in mathematical modeling, signal processing, and any situation requiring oscillatory behavior analysis.

Mathematical Pattern Recognition

Recognizing mathematical patterns, such as periodicity and trigonometric functions within a sequence, is crucial in problem-solving and analysis.

In the given exercise, recognizing that \( \sin(x) \) can serve as a formula for the sequence demands seeing beyond the numbers, understanding their underlying structure. The task involves:

• Identifying repeating cycles in a sequence.
• Associating numerical values with known mathematical functions.
• Generalizing patterns into a formula, such as \( a_n = \sin\left(\frac{\pi}{2} + n\pi\right) \).

Effective pattern recognition allows for efficient solutions and deeper insights into the nature of mathematical sequences. This skill applies across various fields, enhancing our ability to decode, predict, and utilize numerical data seamlessly.