# Concept of Compound Interest

When interest is calculated for every period on the total previous amount, then the total amount of interest gained on all the periods is called compound interest.

Let       Principal = P
Rate of interest = r
No. of years = n

After 1st year, interest     $= {\text{P}} \times {\text{r}}$
After 1st year, amount   $= {\text{P}} + {\text{Pr}}$

Now the principal becomes,

Principal   $= {\text{P}}\left( {1 + {\text{r}}} \right)$
After 2nd year,            interest      $= {\text{P}}\left( {1 + {\text{r}}} \right){\text{r}}$
After 2nd year, Amount    $= {\text{P}}\left( {1 + {\text{r}}} \right) + {\text{P}}\left( {1 + {\text{r}}} \right){\text{r}} = {\text{P}}\left( {1 + {\text{r}}} \right)\left[ {1 + {\text{r}}} \right] = {\text{P}}{\left( {1 + {\text{r}}} \right)^2}$
and so on.

The amount after n-years is

$\boxed{{\text{Compound amount = P}}{{\left( {{\text{1}} + {\text{r}}} \right)}^n}}$

Hence

$\boxed{{\text{Compound interest over }}n – {\text{years = P}}{{\left( {{\text{1}} + {\text{r}}} \right)}^n} – {\text{P}}}$