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Sometimes sequences of numbers are defined recursively, so that given the previous terms of the sequence we can find the next term. A classic example of this is the sequence of Fibonacci numbers, where every two consecutive terms determine the next term, and so if we are given the first two terms, we can calculate the whole sequence. However, if we wanted to, say, calculated the 1000th Fibonacci number, we’d have to start out with the first two, add them to compute the 3rd, add the second and third to compute the 4th, and so on, to slowly build up every single Fibonacci number before the 1000th in order to get there. It’d be much nicer if we had a formula for the Fibonacci sequence. That way, if we wanted the 1000th Fibonacci number, all we’d have to do is plug 1000 into the formula to compute it. Of course, the formula is only useful if it takes less time to crunch it out than it does to do it the brute-force way. A method of finding closed formulas for sequences defined by recurrences is the use of generating functions. Generating functions are functions which “encapsulate” all the information about a sequence, except you can define it without knowing the actual terms of the sequence. The power of generating functions comes from the fact that you can do things like add and multiply them together to create generating functions of other sequences, or write them in terms of themselves to find an explicit formula. Once you have an explicit form for a generating function, you can use some algebra to “extract” the information from the function, which usually means you can find a formula for the sequence in question.

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