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A recursion formula defines each term in a sequence based on previous terms. This means that to find the current term, we must rely on the values that have come before it.
In our original exercise example, the recursion formula is given as \( a_n = (n-1) a_{n-1} + (n-2) a_{n-2} \) for \( n \geq 3 \). Here, the term \( a_n \) is expressed as a combination of the two previous terms: \( a_{n-1} \) and \( a_{n-2} \). This formula incorporates the position \( n \) in the sequence by using the coefficients \( n-1 \) and \( n-2 \).
This kind of formula is a key characteristic of recursive sequences, where each term uses specific rules to relate back to its predecessor(s). Recursive sequences can be powerful because a simple formula can generate complex sequences.

Initial terms in a recursive sequence are crucial because they serve as the starting point for generating the sequence. Without the initial terms, we wouldn't be able to calculate subsequent terms using the recursion formula. In the exercise provided, we have the initial terms \( a_1 = 1 \) and \( a_2 = -1 \). These are essential because all of the following terms depend directly or indirectly on these values.
Think of initial terms as the anchors of the sequence. They are predetermined and set the stage for the entire unfolding of the sequence. As we use the recursion formula to find new terms, we always rely on these initial terms to start the process. Therefore, choosing the right initial terms is crucial for building the sequence according to the problem's requirements.

Sequence calculation using the recursion formula and initial terms involves step-by-step iterations to find each new term. For example, once we have \( a_1 = 1 \) and \( a_2 = -1 \), we apply the recursion formula to find \( a_3 \) and \( a_4 \).
For \( a_3 \):
Plug \( n = 3 \) into the formula: \( a_3 = (3-1)a_2 + (3-2)a_1 \).
Calculate step-by-step: \( a_3 = 2\times(-1) + 1\times1 = -2 + 1 = -1 \).
Now, \( a_3 = -1 \).
Then, for \( a_4 \):
Plug \( n = 4 \) into the formula: \( a_4 = (4-1)a_3 + (4-2)a_2 \).
Calculate step-by-step: \( a_4 = 3\times(-1) + 2\times(-1) = -3 - 2 = -5 \).
So, \( a_4 = -5 \).
Each calculation builds upon the previous terms, showing how the sequence evolves from its starting points. This method highlights the interconnected nature of recursive sequences, where each term is both a result of earlier terms and a foundation for later ones.