Answer:
[tex]P_4 = 7604.08[/tex]
Step-by-step explanation:
If the population increases at a rate of 4% per annum, then:
In year 1:
[tex]P_1 = P_0 + 0.04P_0[/tex]
Where [tex]P_0[/tex] is the initial population and [tex]P_n[/tex] is the population in year n
In year 2
[tex]P_2 = P_1 + 0.04P_1[/tex]
It can also be written as:
[tex]P_2 = P_0 + 0.04P_0 + 0.04 (P_0 + 0.04P_0)[/tex]
Taking out common factor [tex]P_0[/tex]
[tex]P_2 = (1 + 0.04) (P_0) + 0.04P_0 (1 + 0.04)[/tex]
Taking out common factor (1 + 0.04)
[tex]P_2 = (1 + 0.04) (P_0 + 0.04P_0)[/tex]
Taking out again common factor [tex]P_0[/tex]
[tex]P_2 = (1 + 0.04) (1 + 0.04) P_0[/tex]
Simplifying
[tex]P_2 = P_0 (1 + 0.04) ^ 2[/tex]
So
[tex]P_n = P_0 (1 + 0.04) ^ n[/tex]
This is the equation that represents the population for year n
Then, in 4 years, the population will be:
[tex]P_4 = P_0 (1 + 0.04) ^ 4\\P_4 = 6500(1 + 0.04) ^ 4\\P_4 = 7604.08[/tex]
