What is the Compound Interest Formula?
Instead of calculating interest year by year (which is tedious for long periods), we use a direct formula:
Amount Formula:
\[
A = P\left(1 + \frac{r}{100}\right)^n
\]
Compound Interest Formula:
\[
CI = A - P = P\left[\left(1 + \frac{r}{100}\right)^n - 1\right]
\]
Where:
- A = Amount after n years
- P = Principal (initial amount)
- r = Rate of interest per annum (in %)
- n = Number of years
Derivation of Formula (for n = 3):
- A₁ = P(1 + r/100)
- A₂ = A₁(1 + r/100) = P(1 + r/100)²
- A₃ = A₂(1 + r/100) = P(1 + r/100)³
- Therefore, Aₙ = P(1 + r/100)ⁿ
Important Notes:
- The formula assumes interest is compounded annually
- r must be in percentage (not decimal)
- n must be in same time unit as compounding period