What is the formula of Compound Interest (C.I)?

Hint: We start solving the problem by recalling the definitions of simple and compound interests for a given principal amount and the annual rate of interest. We then assume the variables for the rate of interest and principal amount to derive the rate of interest. We find the final amounts that were obtained after adding interests to the principal amount every year and deduce the equation for the final amount after ‘n’ years. We then subtract the principal amount from the final amount to get the total interest paid on it.

Complete step-by-step solution
According to the formula, we need to find the formula of Compound Interest (C.I).
Let us recall the definitions of simple and compound interests.
We know that simple interest (S.I) is the cost of borrowing money, which is based on the principal amount of the loan or deposit.
$\Rightarrow S.I=PTR$ ---(1), where P = principal amount of loan or deposit.
T = Time (in years).
R = Annual rate of interest.
Whereas the compound interest (C.I) is based on the principal amount and the interest that accumulates on it in every period.
Let us assume the principal amount be P, and the time in which money is returned be n years, and the rate of interest r.
Let us find the total interest that is paid on the principal amount in the first year.
So, we get $I=P\times r=\Pr $.
Now, we get the final amount after the first year by adding interest is $A=P+\Pr =P\left( 1+r \right)$ ---(1).
Now, let us find the interest for the amount we obtained after the first year.
So, we get interest as ${{I}_{1}}=P\left( 1+r \right)r$.
Now, we get the final amount after second year by adding interest is $A=P\left( 1+r \right)+P\left( 1+r \right)r=\left( P\left( 1+r \right) \right)\left( 1+r \right)$.
$\Rightarrow A=P{{\left( 1+r \right)}^{2}}$ ---(2).
Now, let us find the interest for the amount we obtained after the second year.
So, we get interest as ${{I}_{2}}=P{{\left( 1+r \right)}^{2}}r$.
Now, we get the final amount after second year by adding interest is$A=P{{\left( 1+r \right)}^{2}}+P{{\left( 1+r \right)}^{2}}r=\left( P{{\left( 1+r \right)}^{2}} \right)\left( 1+r \right)$.
$\Rightarrow A=P{{\left( 1+r \right)}^{3}}$ ---(3).
From equations (1), (2), and (3), we can see that the final amount we get after ‘n’ years resembles the form $A=P{{\left( 1+r \right)}^{n}}$.
Now, let us find the total interest paid on the principal amount after n years.
So, we get $C.I=P{{\left( 1+r \right)}^{n}}-P=P\left( {{\left( 1+r \right)}^{n}}-1 \right)$.
So, we have found compound interest as $C.I=P\left( {{\left( 1+r \right)}^{n}}-1 \right)$.

Note: We should not stop solving the problem after finding the final amount after ‘n’ years as it contains both the principal amount and the interest that is paid on the principal amount every year. We should know that simple interest is common every year which is not in the case of compound interest (compound interest increases every year). Similarly, we can also expect problems to find the compound interest for a given principal amount and the annual rate of interest.