Problem 2

Question

Determine if the given sequence is increasing, decreasing, or not monotonic.$\\{\sin n \pi\\}$

Step-by-Step Solution

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Answer

The sequence is not monotonic.

1Step 1: Understand the Sequence

The given sequence is $ \sin(n \pi) $, where $n$ is an integer. This sequence will need to be analyzed for its behavior.

2Step 2: Evaluate the Sine Function

Recall that $ \sin(n \pi) = 0 $ for any integer value of $n$. This is because sine of any multiple of $ \pi $ is zero.

3Step 3: Check for Monotonicity

A sequence is increasing if each term is greater than or equal to the previous term. It is decreasing if each term is less than or equal to the previous term. For the given sequence, each term is zero: $0, 0, 0, \ldots$.

4Step 4: Conclusion

Since each term in the sequence $ \sin(n \pi) $ is constant and equal to zero, it is neither increasing nor decreasing.

Key Concepts

Monotonic SequenceSine FunctionInteger Sequence

Monotonic Sequence

In mathematics, a **monotonic sequence** is one that either increases or decreases consistently. It does not fluctuate between values. A sequence is called:

li>**Increasing** if each term is greater than or equal to the previous term.

**Decreasing** if each term is less than or equal to the previous term.

An **increasing sequence** looks like this: 1, 2, 3, 4, ...A **decreasing sequence** looks like this: 4, 3, 2, 1, ...
The key here is consistency in trend - either always going up or always going down.
Now let's consider our original sequence: $\sin(n \pi )$. Each term, regardless of $n$, is zero: $0, 0, 0, \ldots$. Since it doesn’t consistently increase or decrease, it’s not monotonic.

Sine Function

The **sine function** is a trigonometric function represented as $ \sin(x) $. It describes a smooth, periodic oscillation. Here are some key points:

$ \sin( \pi) = 0 $ and $\sin(2 \pi) = 0 $.
$ \sin(3 \pi) = 0 $
In fact, $\sin(n \pi) = 0$ for any integer $n$.

The sine function has a period of $2 \pi$, meaning it repeats its pattern every $2 \pi$ units. For our sequence, this periodicity plays a key role. Because $ \sin(n \pi) $ results in zero for any integer $n$, each term in the sequence is zero: $0, 0, 0, \ldots$. So, our sequence does not change as $n$ changes.

Integer Sequence

An **integer sequence** $\{a_n\}$ is a sequence where each term $a_n$ is an integer. Examples are $0, 1, 2, \ldots$ or $2, 4, 6, \ldots$. In our specific case, we look at $\sin(n \pi)$, where $n$ is an integer.

Since $n$ is an integer, $\sin(n \pi)$ corresponds to sine values for multiples of $ \pi $.

Remember, sine of any integer multiple of $ \pi $ is zero. Therefore, our sequence is:

$0, 0, 0, \ldots$.

Each term being zero satisfies the integer sequence property that the terms are integers. But unlike more interesting integer sequences that vary, this one remains constant.

Problem 2

Problem 3

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