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In this paragraph:
"Now imagine that there are $x_ n$ pairs of rabbits after $n$ months. The number of pairs in month $n+1$ will be $x_ n$ (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be $x_{n-1}$ new pairs."
You say that new pairs are only born to pairs at least 1 month old but is this correct? Earlier you state that pairs can mate at 1 month but only produce new pairs at 2 months. Therefor new pairs are only born to pairs at least 2 months old (not 1 month old). If this is so, how does it affect your formula?
Thanks