### 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:

**: Population growth after \(n\) years:**

*Type 1*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 $A=P\phantom{\rule{0.147em}{0ex}}{\left(1+\frac{r}{100}\right)}^{n}$.

**: Population grows at different rates:**

*Type 2*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.