Step-by-Step Derivation of the Compund Interest Formula

The formula is derived by understanding that each period's interest is added to the principal, which then earns interest in the next period.

Let's first assume compounding happens once per year (n=1). The rate is just r.

• At the end of Year 1: You have your principal plus the interest. 
  A1 = P + (P * r) = P(1 + r)
• At the end of Year 2: Your new principal is A1. You earn interest on this new, larger amount. 
  A2 = A1(1 + r) = [P(1 + r)](1 + r) = P(1 + r)2
• At the end of Year 3: Your new principal is A2. 
  A3 = A2(1 + r) = [P(1 + r)2](1 + r) = P(1 + r)3

Seeing the pattern, for 't' years, the formula is A = P(1 + r)t.

Now, let's introduce 'n' compounding periods. Instead of compounding once per year at rate r, we compound 'n' times per year. This means:

1. The interest rate per period is the annual rate divided by the number of periods: rate per period = r/n.
2. The total number of compounding periods is the number of years multiplied by the periods per year: total periods = n * t.

By substituting these new values into our simple formula A = P(1 + rate)periods, we get the final compound interest formula: