 # Apply, or unroll, the definition

Several rational equations, involving factorials, can be solved by unrolling the definition, then cancelling the common factors. We can see that this technique of unrolling a recursive definition can be used beyond factorials, although I've yet to see the skill taught explicitly. For example, let $$T_n$$ be the $$n$$-th triangular number. We can solve the following equation by unrolling:

\begin{align} & T_{n + 3} - T_{n + 1} = 9 \\[0.5em] & (n + 3) + (n + 2) + T_{n + 1} - T_{n + 1} = 9 \\[0.5em] & 2n + 5 = 9 \\[0.5em] & n = 2 \end{align}

My preference is to say "unroll" when referring to the use of a recursive definition, and say "apply" when referring to a static definition. We find applying, or unrolling, the definition of a function, is used to solve a wide variety of problems. As an example of applying a definition, we often do this when solving absolute value equations. We use the rule $$\lvert x \rvert = y$$ implies $$-x = y$$ or $$x = y.$$

The technique of applying a definition is used throughout mathematics, not just at the elementary level. For example, to prove basic facts about the integers, we have to define a congruence on pairs of natural numbers. Every time we want to prove something about integers, we have to unroll this congruence definition in order to do so.