Why is there a constant 1 in the compound interest formula?

Question

(This is about my calculus project) I am curious about why there is a 1 in the compound interest formula. Like what does the 1 do in the function?

$$A(t)= P (1+\dfrac rn)^{nt}$$

Answer 1

If the interest rate is $p$ percent, then that's a factor of $\frac p{100}$, which means that your interest on amount $A_0$ is $\frac p{100}\cdot A_0$. So after adding the interest amount to your base amount your new total amount is $$A_1=A_0+\frac p{100}A_0=\left(\color{red}1+\frac p{100}\right)\cdot A_0.$$ If you repeat this ("multiply current amount by $1+\frac p{100}$ to get new amount") $n$ time, you arrive at $$A_n=\left(\color{red}1+\frac p{100}\right)^n\cdot A_0.$$ So the $\color{red}1$ ultimately comes from the fact that you keep your base amount and not only the smaller interest amount.

Answer 2

It's there to ensure that the end result is at least a 1 so that if you multiply it to your principle you get no less than what you started with as principle. This works for any rate r that's 0 or more. Otherwise you would get a final amount that can be less than what your principle started with originally and that would be surreal. This is like saying that it's possible you can lose money when the compounding rate is 0 or greater which means you got robbed, shouldn't be possible.