Question

The population of a town is 6500 and is increasing at a rate of 4% each year. Create a function for this scenario. At this rate, approximately what will the town's population be in 4 years?

Answer 1

`P_4 = 7604.08`

Explanation:

If the population increases at a rate of 4% per annum, then:

In year 1:

`P_1 = P_0 + 0.04P_0`

Where
`P_0` is the initial population and
`P_n` is the population in year n

In year 2

`P_2 = P_1 + 0.04P_1`

It can also be written as:

`P_2 = P_0 + 0.04P_0 + 0.04 (P_0 + 0.04P_0)`

Taking out common factor
`P_0`

`P_2 = (1 + 0.04) (P_0) + 0.04P_0 (1 + 0.04)`

Taking out common factor (1 + 0.04)

`P_2 = (1 + 0.04) (P_0 + 0.04P_0)`

Taking out again common factor
`P_0`

`P_2 = (1 + 0.04) (1 + 0.04) P_0`

Simplifying

`P_2 = P_0 (1 + 0.04) ^ 2`

So

`P_n = P_0 (1 + 0.04) ^ n`

This is the equation that represents the population for year n

Then, in 4 years, the population will be:

`P_4 = P_0 (1 + 0.04) ^ 4\\P_4 = 6500(1 + 0.04) ^ 4\\P_4 = 7604.08`