So far, in  Simple InterestIntroduction To Simple Interest,  we have seen that interest is calculated on a fixed amount for whatever time period that is specified. However, in real life most of the interest calculations happen in a slightly different way.
When money is deposited in the bank or somebody borrows money from the the bank, the interest for the first period is added to the principal amount. This gives the amount for calculating the interest for the second period. The same thing repeats over every period, that is, the interest for the current period is calculated on the sum of the principal and interest of the previous period.

Suppose you deposit an amount $P$ with a bank that pays $R\%$ interest per annum.

So, at the end of the first year your account has a total of $P + \dfrac{R \times P}{100} = P\left(1 + \dfrac{R}{100}\right)$

Let us say you do not withdraw any money from your account, so the second year's interest will be calculated on the new principal amount,  $P\left(1 + \dfrac{R}{100}\right)$

So the interest for the second year will be .

$\texttip{\therefore}{therefore}$ At the end of second year the total amount in your account will be:

$P\left(1 + \dfrac{R}{100}\right) + P\left(1 + \dfrac{R}{100}\right) \times \dfrac{R}{100}$

Taking $P\left(1+\dfrac{R}{100}\right)$ common in the above expression we get:

$P\left(1 + \dfrac{R}{100}\right)\left(1+\dfrac{R}{100}\right) = P\left(1+\dfrac{R}{100}\right)^2$

Similarly, at the end of $3$ years the total amount in your account will be:

$P\left(1+\dfrac{R}{100}\right)^3$

and at the end of $N$ years, the total amount in your account will be:

$A = P\left(1 + \dfrac{R}{100}\right)^N$

where,

$P = the\ principal\ amount,$

$R = the\ annual\ rate\ of\ interest\ in\ \%,$

$N = the\ time\ period\ in\ years,\ and$

$A = the\ total\ amount\ at\ the\ end\ of\ N\ years$

Now let us try the following example:

Remember, this formula is applicable for interests that are compounded annually only. There are cases where interests are compounded daily, monthly, quarterly, and in some cases continuously for extremely small time intervals. These cases use different formulas for calculating the amount or interest. We will see those cases later.