The population of a town increases every year by 2% of the population at the beginning of that year. The number of years by which the total increase of population be 40% is:
To determine the number of years by which the total increase in population is 40%, we need to set up an equation and solve for the unknown variable.
Let's assume the population at the beginning of the year is P.
The increase in population after the first year would be 2% of P, which is 0.02P.
So, after one year, the population would be P + 0.02P = 1.02P.
Similarly, after the second year, the population would be 1.02P + 0.02(1.02P) = 1.02P + 0.0204P = 1.0404P.
We can see that after each year, the population is increasing by 2% of the previous year's population.
To find the number of years it takes for the total population increase to be 40%, we can set up the following equation:
P + 0.02P + 1.02P + 1.0404P + ... = P + 0.4P
Simplifying the equation, we get:
P(1 + 0.02 + 1.02 + 1.0404 + ...) = P(1.4)
Cancelling out the P on both sides, we have:
1 + 0.02 + 1.02 + 1.0404 + ... = 1.4
Now, we need to find the sum of an infinite geometric series with a common ratio of 1.02.
The sum of an infinite geometric series is given by the formula:
S = a / (1 - r)
Where S is the sum of the series, a is the first term, and r is the common ratio.
Plugging in the values, we have:
S = 1 / (1 - 1.02)
Simplifying the expression, we get:
S = 1 / (-0.02)
S = -50
Since the sum of the series is -50, this means that the equation does not have a finite solution. It implies that the population will never reach a total increase of 40%.
Therefore, the number of years by which the total increase in population can be 40% is infinite.