Question

The principal Po is invested in an account that pays an annual interest rate r (written as a decimal), compounded n times per year.
The formula for the amount of money in the account at the end of the first period is given by the formula: P=Po(1+ r/n).

Explain why the amount of money in the account at the end of the first year is given by the formula with n as the exponent: P=Po(1+ r/n)^n

[o=0 placed at the bottom of the number]

Answer 1

At the end of the first year, the correct formula is this one:
$P=P_0(1+r)$
The second year, we add r times the above amount, we get:
$P_0(1+r)+r(P_0(1+r))=P_0(1+2r+r^2)\\=P_0(1+r)^2$
Iterating the process, we get the general formula of any number of years:
$P=P_0(1+r)^n$