Derivation of the Compound Interest Formula Without Using Calculus

Question

I am a student but I am only in Algebra 2. I am doing a project on compound interest and I am trying to prove how we arrive at the formula $y=Pe^{rt}$ but I am stuck at the step where you get from $y=P(1+r/n)^{nt}$ to $y=P((1+1/n)^{n/r})^{rt}$. I would really appreciate some help with my algebra. Thanks!

Answer 1

Generally, $y=P(1+r/n)^{nt}$ does not imply $y=P((1+1/n)^{n/r})^{rt}$ (Try $P=1, n=2, r=2, t=2$. It fails, since $P(1+r/n)^{nt}=1(1+\frac{2}{2})^{2 \cdot 2}=16 \neq P((1+1/n)^{n/r})^{rt}=1((1+\frac{1}{2})^{2/2})^{2\cdot 2}=\frac{81}{16})$.

It looks like something may have been mistyped here.

However, it is true that $$y=P(1+r/n)^{nt}=P((1+r/n)^{n/r})^{rt}.$$

There is a rule for exponentiation that states $(a^b)^c=a^{bc}$. Assuming that $r$ is nonzero, we can write $$y=P(1+r/n)^{nt}=P(1+r/n)^{\frac{n}{r}\cdot (rt)}=P((1+r/n)^{\frac{n}{r}})^{\cdot rt}.$$