In the $1^{st}$ month, a farm has a pair of rabbits. Then in the $3^{rd}$ month, the pair of rab

Question

In the $1^{st}$ month, a farm has a pair of rabbits. Then in the $3^{rd}$ month, the pair of rabbits grow up and reproduce another pair of rabbits. Suppose that: 1. It always takes two months for rabbits to grow up and be able to reproduce baby rabbits. 2. After growing up, every pair of rabbits reproduce one pair of rabbits every month. 3. No rabbit dies on this farm. Find the number of pair(s) of rabbits on the farm with respect to time (in month), by completing the following table. (Hint: from the $3^{\text{rd}}$ month onwards, the number of pair(s) of rabbits in each month equals the sum of the number of pair(s) of rabbits in the previous month and the number of pair(s) of new-born rabbits this month.) Time (Month)Number of Pair(s) of Rabbits, $P_n$ $1$$1$ $2$$1$ $3$$2$ $4$$3$ $5$$5$ $6$$8$ $7$ $8$ $9$ State a rule of the number sequence formed by the number of pair(s) of rabbits on the farm, $P_n$. Based on your answers in (i) and (ii), Write down the next two terms of the following sequences. $-3, 2, -1, 1, 0, \dots$ $3, 2, 5, 7, 12, \dots$

Accepted Answer

(i) $13,21,34$ (ii) $P_1=P_2=1$ and $P_n=P_{n-1}+P_{n-2}$ for $n\ge3$ (iii)(a) $1,1$ (iii)(b) $19,31$

Time (Month)	Number of Pair(s) of Rabbits, \(P_n\)
\(1\)	\(1\)
\(2\)	\(1\)
\(3\)	\(2\)
\(4\)	\(3\)
\(5\)	\(5\)
\(6\)	\(8\)
\(7\)
\(8\)
\(9\)

Number Patterns

Number Patterns

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