Answer:
Let's start by finding out how many new mosquitos enter the room every hour. We know that the number triples every hour, so the sequence of new mosquitos entering the room would be:
1, 3, 9, 27, 81, ...
We can see that this is a geometric sequence with a common ratio of 3 and a first term of 1. So the formula for the nth term of this sequence is:
a_n = a_1 * r^(n-1)
where a_1 is the first term, r is the common ratio, and n is the number of terms.
So, for example, after 3 hours, the number of new mosquitos that have entered the room would be:
a_3 = 1 * 3^(3-1) = 9
Now let's use this formula to find the total quantity of mosquitos that have entered the room in 12 hours. We can do this by summing up the first 12 terms of the sequence:
S_12 = a_1 * (r^12 - 1) / (r - 1)
where S_12 is the sum of the first 12 terms.
Plugging in the values, we get:
S_12 = 1 * (3^12 - 1) / (3 - 1) = 88573
Therefore, the total quantity of mosquitos that have entered the room in 12 hours is 88,573.