Hi there -
So an exponential function is a function that describes a situation where you are multiplying something by the same number over and over again, like the population increase problem you present here. After 1 year, the population will grow by:
48500 * 2.5% =
48500 * 0.025 = 1212.5 (or 1213.. no half people!!)
for a total of 49713 people
another way of looking at that is that the total population after one year, will by 102.5% of the population the year before:
48500 * 102.5% =
48500 * 1.025 = 49,712.5 (or 49713, rounded to the nearest whole person)
the following year the population will grow by another 2.5%, but this time its 2.5% of 49,712.5, not 48,500.
so that's kind of like:
49,712.5 * 1.205
48500 * 1.025 * 1.025
notice that for 2 years, we have multiplied by 1.025 TWO times. For 3 years, we'd multiply by 1.025 three times. For 13 years, 13 times... and so on.
We express this mathematically like this:
Starting population * growth rate plus 100% raised to the power of the number of years
For this example:
to find out what the population is after 8 years, we just plug in 8 for 'n'
48500 (1.0258)= approx. 59,092.54
rounded to the nearest hundred : 59,100