Question

Determine whether the sequence is monotonically increasing or decreasing. If it is not, determine if there is an \(m\) such that it is monotonic for all \(n \geq m\). $$\left\\{a_{n}\right\\}=\left\\{\frac{n}{n+2}\right\\}$$

Short answer

The sequence is monotonically increasing for all \( n \geq 1 \).

Step-by-step solution

Understand the Sequence

We are given the sequence \( \{ a_n \} = \{ \frac{n}{n+2} \} \). This means for each positive integer \( n \), the term of the sequence is calculated as \( a_n = \frac{n}{n+2} \).

Analyze Monotonicity

To determine if the sequence is monotonically increasing, we check the difference between consecutive terms. Calculate \( a_{n+1} - a_n \): \[ a_{n+1} = \frac{n+1}{n+3} \] \[ a_n = \frac{n}{n+2} \]Thus, \[ a_{n+1} - a_n = \frac{n+1}{n+3} - \frac{n}{n+2} \]Simplifying this expression will reveal whether the sequence increases or decreases.

Simplifying the Difference

Compute the difference \( a_{n+1} - a_n \):\[ a_{n+1} - a_n = \frac{n+1}{n+3} - \frac{n}{n+2} \]Combine this into a single fraction:\[ = \frac{(n+1)(n+2) - n(n+3)}{(n+3)(n+2)} \] \[ = \frac{n^2 + 2n + n + 2 - (n^2 + 3n)}{n^2 + 5n + 6} \] \[ = \frac{n + 2}{n^2 + 5n + 6} \]Checking the numerator, since \(n+2\) is always positive for all \(n \geq 1\), the difference is positive. Thus, the sequence is increasing.

Conclusion on Monotonicity

The analysis shows \( a_{n+1} > a_n \) for all \( n \geq 1 \). The sequence \( \{ a_{n} = \frac{n}{n+2} \} \) is monotonically increasing for all positive integers \( n \). No further check for \( m \) is needed, because it is already increasing for all terms.

Key concepts

Monotonicity

Monotonicity in sequences refers to the consistent behavior of a sequence in either increasing or decreasing. In simpler terms, a sequence is considered monotonic if it does not change its direction of movement. This means all its terms either go up in a continuous manner or go down without switching back and forth.

To identify if a sequence is monotonic, you can calculate the difference between consecutive terms and see if there's a consistent pattern. If a sequence is increasing or decreasing entirely or from a specific term onwards, it helps in classifying its nature effectively.

A sequence that rises or falls steadily provides insights into its behavior, which can be crucial for mathematical predictions and analysis.

Sequence Analysis

Sequence analysis involves examining the structure and behavior of a sequence to determine its characteristics. By systematically inspecting the terms, their relationships, and how they change from one to the next, one can gather insights about its general behavior.

The given sequence, \( \{ \frac{n}{n+2} \} \), is an example where analysis plays a key role in understanding its direction. Through analysis, we aim to discover whether it maintains a specific pattern, in this case, whether it's increasing as \( n \) increases.

Analyzing involves often breaking down the sequence components and utilizing mathematical operations such as subtraction (finding the difference in terms) or calculus for more complex sequences. This detailed approach sheds light on the monotonic nature and any underlying properties, such as limits or convergence points.

Difference of Terms

The difference between terms in a sequence offers valuable information about its behavior. By calculating \( a_{n+1} - a_n \), which represents the change between consecutive terms, we can determine if the sequence is increasing or decreasing.

For the sequence \( a_n = \frac{n}{n+2} \), the difference \( \frac{n+1}{n+3} - \frac{n}{n+2} \) simplifies to \( \frac{n + 2}{n^2 + 5n + 6} \). This expression reveals that because the numerator \( n+2 \) is always positive for all \( n \geq 1 \), the entire difference ends up being positive as well.

Thus, the difference of terms shows that as \( n \) increases, the sequence does not decrease; instead, it consistently increases. Understanding this difference is a key step to determine the overall monotonicity of the sequence.

Increasing Sequence

An increasing sequence is one where each term is greater than the preceding term. This reflects a steady rise in the values as you move through the sequence.

From our analysis of \( \{ \frac{n}{n+2} \} \), we demonstrated that for every \( n \), the term \( a_{n+1} \) is always greater than \( a_n \). That means that the entire sequence is monotonically increasing starting from the very first term.

This behavior is confirmed by our earlier calculation of the difference \( a_{n+1} - a_n \), which turned out positive. Since the increment between the terms is never negative, you can confidently state that this sequence is indeed increasing. Recognizing this is valuable in mathematical analysis, often leading to conclusions about limits or convergence as \( n \) approaches infinity.