Answer
See explanation
Work Step by Step
**Solution Explanation**
We have a single pair of rabbits (one male–female pair) born at the very beginning of month 0. They are **not** fertile for their first two months of life, but at the end of every month **thereafter** they produce one **new** male–female pair. No rabbits ever die.
Let \(s_n\) be the number of pairs of rabbits **alive at the end of month \(n\)**. We are told \(s_0 = 1\) (the original newborn pair at time 0). We wish to:
1. Find a recurrence relation for \(s_n\).
2. Compute \(s_0, s_1, s_2, s_3, s_4, s_5\).
3. Determine how many **rabbits** there are at the end of 12 months (i.e., at the end of month 12).
---
## (a) Recurrence Relation
- **At the end of month 0:** \(s_0 = 1\).
- **At the end of month 1:** The original pair is only 1 month old (not yet fertile), so no new pairs are born. Thus \(s_1 = 1\).
- **At the end of month 2:** The original pair is now exactly 2 months old—but the condition says they are **not** fertile during their first two months, so they still do **not** reproduce. Hence \(s_2 = 1\).
For **\(n \ge 3\)**, any pair that is **at least** 2 months old will produce a new pair at the end of the month.
- The number of pairs that were alive **two months earlier** (i.e.\ at the end of month \(n-2\)) is \(s_{n-2}\).
- Exactly those \(s_{n-2}\) pairs are now old enough to reproduce at the end of month \(n\).
- Each of those pairs produces **1** new pair.
Hence the total number of pairs at the end of month \(n\) is
\[
\text{(old pairs)} + \text{(new pairs)}
\;=\;
s_{n-1} \;+\; s_{n-2}.
\]
Therefore, the recurrence is
\[
\boxed{s_0 = 1,\quad s_1 = 1,\quad s_2 = 1,\quad \text{and for }n \ge 3:\; s_n = s_{n-1} + s_{n-2}.}
\]
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## (b) Compute \(s_0, s_1, s_2, s_3, s_4, s_5\)
Using the initial conditions and the recurrence \(s_n = s_{n-1} + s_{n-2}\) for \(n\ge3\):
1. \(s_0 = 1.\)
2. \(s_1 = 1.\)
3. \(s_2 = 1.\)
4. \(s_3 = s_2 + s_1 = 1 + 1 = 2.\)
5. \(s_4 = s_3 + s_2 = 2 + 1 = 3.\)
6. \(s_5 = s_4 + s_3 = 3 + 2 = 5.\)
So the first six values (\(n=0\) through \(5\)) are
\[
s_0=1,\; s_1=1,\; s_2=1,\; s_3=2,\; s_4=3,\; s_5=5.
\]
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## (c) How Many Rabbits at the End of the Year?
A “year” here is 12 months, so we want \(s_{12}\), the number of **pairs** at the end of month 12. By continuing the same Fibonacci‐type recurrence, one finds
\[
s_6=8,\; s_7=13,\; s_8=21,\; s_9=34,\; s_{10}=55,\; s_{11}=89,\; s_{12}=144.
\]
Thus there are \(s_{12} = 144\) **pairs** of rabbits at the end of month 12. Since each pair has 2 rabbits, the total number of individual rabbits is
\[
2 \times s_{12} \;=\; 2 \times 144 \;=\; 288.
\]
Hence, \(\boxed{288}\) rabbits in total at the end of the year.
