Final answer:
The recurring relation for the number of bats is Bn = 1.5 × Bn-1, where B1 = 1200. We solve this exponential growth pattern with Bn = 1200 × (1.5)n-1. Applying to the 12th count, we get that the bat population would be approximately 99144 or 99145, depending on rounding.
Step-by-step explanation:
To solve this problem, we will first establish a recurrence relation for the number of bats. Observing the pattern of growth in the counts (1200, 1800, 2700, 4050), it's clear the bats are increasing by a factor of 1.5 every 2 months (since 1800 is 1.5 times 1200, 2700 is 1.5 times 1800, and so on).
Let's denote the number of bats at the nth count as Bn. Hence the recurrence relation can be defined as Bn = 1.5 × Bn-1 with the initial count B1 = 1200.
To solve the relation, recognizing it as an exponential growth problem, we can use the closed form for the initial term multiplied by the growth rate raised to the (n-1) power. Therefore, the solution would be Bn = 1200 × (1.5)n-1.
For the 12th count, we substitute n with 12 in the formula to get B12 = 1200 × (1.5)11 because the counts started from B1. After calculating, we find that B12 is 99144.50. However, since you cannot have half a bat, the population would be 99144 or 99145 depending on how you round.