Rabbit Problem/Solution

Solution
At the end of the first month, the initial pair will produce $1$ new pair of rabbits, making a total of $2$ pairs.

At the end of the second month, the initial pair will produce $1$ more new pair, making a total of $3$.

At the end of the third month:
 * the initial pair will produce $1$ more new pair
 * the second pair will produce $1$ pair, as they are now able to breed.
 * This results in $5$ pairs.

At the end of month $n$, the total count of rabbit pairs is:
 * the count of the rabbits which were alive at the end of the previous month (all are still alive)

plus:
 * the count of the rabbits which were alive at the end of the month before that (as all those are of an age to breed).

Let $R \left({n}\right)$ be the number of pairs of rabbits at the end of month $n$.

Thus, by the above reasoning:
 * $R \left({0}\right) = 1$
 * $R \left({1}\right) = 2$
 * $R \left({2}\right) = 3$
 * $R \left({n}\right) = R \left({n - 1}\right) + R \left({n - 2}\right)$

where $R \left({0}\right)$ is understood as being the number of pairs of rabbits at the end of month $0$, that is, the beginning of month $1$.

Thus it is seen that:
 * $R \left({n}\right) = F_{n + 2}$

where $F_{n + 2}$ denotes the $n + 2$th Fibonacci number.

Hence, at the end of $12$ months, there will be $F_{14} = 377$ rabbits.