Groups: It is important to talk about Problem D2 as a whole group, as this question underscores the importance and convenience of the recursive formula. If you know the number of stairs in the nth staircase, the number of stairs in the next staircase can be found by adding n + 1 onto the previous total.
Notice that the recursive formula can be applied only in situations where you know the previous term. Part of our job is to determine in which situations different kinds of representations are most useful. What makes a rule useful is how easy it is to apply -- closed-form rules often win the battle here -- and how easy it is to come up with. In the staircase problem, the recursive rule is much easier to come up with than the closed form.
Another example where the recursive rule is easier to find than the closed-form rule is the famous Fibonacci number sequence: 1, 1, 2, 3, 5, 8, 13, ... .
The recursive rule for the sequence is elegant: Start with 1, 1. Then, to get any term in the sequence, add the previous two terms.
The closed-form rule is complicated:
It's not clear that this function produces integer outputs, much less that the outputs are the terms of the sequence above. Imagine trying to come up with such a formula!
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