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Understanding if a sequence is increasing is a fundamental concept in mathematics. A sequence is considered increasing if each term is larger than the term before it. Formally, a sequence \( a_n \) is increasing if \( a_{n+1} > a_n \) for all \( n \) in its domain.

In our exercise, we're asked to determine whether the sequence defined by \( v_n = n! + 2 \) is increasing. To make this determination, you compare consecutive terms of the sequence. If for every \( n \) the inequality \( v_{n+1} > v_n \) holds true, then \( v \) is an increasing sequence. This property of sequences is essential for understanding growth patterns and is also valuable in calculus, specifically when dealing with series convergence.

Factorial notation is used to describe the product of an integer and all the non-zero integers below it. It is denoted by an exclamation point (!). For any positive integer \( n \) the factorial, denoted as \( n! \) is defined as:\[ n! = n \times (n - 1) \times \.\.\. \times 3 \times 2 \times 1 \.\] For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \.\) A special case is \( 0! \) which is defined to be 1.

Factorial notation is not limited to multiplication; it is also connected to permutations and combinations in probability theory, and is even related to many functions and structures within discrete mathematics and combinatorics. In our original exercise, the factorial operator provides a method to create a sequence by using the index \( n \) of each term to generate a value.

Inequalities play a vital role in understanding sequences. They provide a way to compare the size of numbers and, by extension, the terms in a sequence. When dealing with sequences, inequalities tell us whether the sequence is increasing, decreasing, or constant with respect to its terms. In the context of our original problem, the inequality \( (n+1)! + 2 > n! + 2 \) simplifies to \( (n+1) \cdot n! > n! \) after subtracting 2 from both sides.

This simplified inequality directly utilizes the definition of factorial notation \( (n+1)! = (n+1) \cdot n! \) to show that every subsequent term in the sequence is greater than the preceding term, confirming that the sequence is indeed increasing. It is a clear example of how understanding factorial notation and inequalities can be used together to analyze the behavior of a sequence.