Let's solve each question step by step:
1) **Pairs in one year**:
At the end of the first month, we have 1 pair.
At the end of the second month, we still have 1 pair.
At the end of the third month, the original pair gives birth to a new pair, so we have 2 pairs.
At the end of the fourth month, we have 3 pairs (the original pair, plus the new pair).
At the end of the fifth month, we have 5 pairs.
At the end of the sixth month, we have 8 pairs.
This pattern follows the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...
In one year (12 months), we sum the pairs for each month up to the 12th month:
1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144 = 377 pairs.
2) **Pairs in three years**:
To find the number of pairs in three years, we continue the Fibonacci sequence until we reach the 36th month (3 years).
The 36th month in the Fibonacci sequence is 149, so there will be 149 pairs of rabbits after three years.
3) **Time to get more than 10 billion pairs**:
We need to find the month in which the number of pairs exceeds 10 billion.
The Fibonacci sequence doesn't increase linearly; it grows exponentially.
So, we can iterate through the Fibonacci sequence until we find a number exceeding 10 billion.
We need to calculate the Fibonacci sequence iteratively until we reach a number greater than 10 billion.
Once we find the first Fibonacci number greater than 10 billion, we note the month at which it occurs.
