### Theory:

In our day to day life, we observe that there are some situations or things that increase in magnitude over a period of time.

For example:

• The population of a state/country.
• The value of the property.
• The growth of the cell.
• The depreciation in the values of machines, vehicles, etc.,

Similar to the compound interest, the things  mentioned above will also increase over a period.

Let us learn how the compound interest formula is used to calculate these things.
1. Increase in Growth:
Type 1: Population growth after $$n$$ years:
Let $$P$$ be the population of a city or state at the beginning of a certain year, and the population grows at a constant rate of $$r \%$$ per annum.

Then, the population after $$n$$ years is given by $A=P\phantom{\rule{0.147em}{0ex}}{\left(1+\frac{r}{100}\right)}^{n}$.
Type 2: Population grows at different rates:
Let $$P$$ be the population of a city or state at the beginning of a certain year, and the population grows at a constant rate of $$r_{1} \%$$ in the first year, $$r_{2} \%$$ in the second year and so on.

Then, the population after $$n$$ years is given by:

$A=P\left(1+\frac{{r}_{1}}{100}\right)\left(1+\frac{{r}_{2}}{100}\right)\left(1+\frac{{r}_{3}}{100}\right)......\left(1+\frac{{r}_{n}}{100}\right)$
2. Depreciation:
Let $$P$$ be the value of the product or an article at a certain time, and the value of the product depreciates at the rate of $$r \%$$ per annum.

Then, the depreciated value at $$n$$ years is given by $A=P\phantom{\rule{0.147em}{0ex}}{\left(1-\frac{r}{100}\right)}^{n}\phantom{\rule{0.147em}{0ex}}$.
Important!
We can also use the above formulae to calculate the growth of cells in a particular period at a particular rate and to calculatethe value of land increased or decreased at a particular time at a particular rate.