Answer:
Explanation:
(a) Let P(n) denote the population of the town at the start of year n.
The percentage change in population due to births and deaths is given by:
(0.09 - 0.005) = 0.085 or 8.5% per annum.
So, the population at the start of year (n+1) will be:
P(n+1) = P(n) + 0.085P(n)
In addition, a constant number of 250 new residents move into the town each year. Therefore, the population at the start of year (n+1) will be:
P(n+1) = P(n) + 0.085P(n) + 250
Combining these two equations, we obtain the desired recurrence relation:
P(n+1) = 1.085P(n) + 250
(b) To find a formula for the population of the town at the start of the nth year, we can use the recurrence relation and iterate it n-1 times. We have:
P(1) = 3000 (given)
P(2) = 1.085(3000) + 250 = 3385
P(3) = 1.085(3385) + 250 = 3802.25
P(4) = 1.085(3802.25) + 250 = 4262.36
and so on.
Thus, the population of the town at the start of the nth year is given by the formula:
P(n) = (3000 + 250/0.085) * 1.085^(n-1) - 250/0.085