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A sequence formula is a mathematical expression that defines the rule for finding the terms in a sequence. In our context, the sequence is given by the formula \( a_n = \frac{n}{n+2} \). This formula tells us how each term in the sequence is related to its position \( n \), where \( n \) starts at 1 and continues indefinitely.

To find the first five terms, substitute \( n = 1, 2, 3, 4, 5 \) into the formula. It gives us the sequence:

• \( \frac{1}{3} \)
• \( \frac{1}{2} \)
• \( \frac{3}{5} \)
• \( \frac{2}{3} \)
• \( \frac{5}{7} \)

The formula helps us easily identify not just the initial terms, but any term in the sequence by plugging in the appropriate value for \( n \). There's consistency in using the formula, making it a powerful tool in sequence analysis.

The limit of a sequence refers to the value that the terms of a sequence approach as the index \( n \) (representing each term's position) goes to infinity. For the sequence \( a_n = \frac{n}{n+2} \), we want to explore what happens as \( n \) becomes very large.

To do this, rewrite the expression \( \frac{n}{n+2} \) by dividing both the numerator and denominator by \( n \), yielding:\[\frac{1}{1 + \frac{2}{n}}\]
As \( n \to \infty \), the term \( \frac{2}{n} \) becomes negligibly small. Thus, the expression simplifies to \( \frac{1}{1} = 1 \).

This shows that the sequence converges to the limit of 1. In simpler terms, the larger \( n \) becomes, the closer each term in the sequence gets to the value 1.

An increasing sequence is a sequence where each successive term is larger than the previous one. Our sequence \( a_n = \frac{n}{n+2} \) demonstrates this characteristic when we observe the progression of terms like \( \frac{1}{3}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{5}{7} \).

To analyze whether this sequence is increasing:

• Calculate the difference between successive terms: \( a_{n+1} - a_n \).
• Check if the difference is positive, which implies an increase.

Calculating this difference can be formalized as: \[ \frac{n+1}{n+3} - \frac{n}{n+2} \]. If this expression is positive for all \( n \geq 1 \), the sequence is confirmed to be increasing.

In practical terms, our already computed sequence terms show an evident increase, indicating that \( \frac{n}{n+2} \) is indeed an increasing sequence as \( n \) grows. This property is essential since it assists in predicting the behavior of a sequence (e.g., convergence towards a limit).