A problem from FM exam on accumulation and amount function, please help!

Question

Suppose that accumulation function $a(t)$ is differentiable and satisfies $$a(s+t)=a(s)a(t)$$

for all nonnegative real numbers s and t.

$a'(t) = a'(0)a(t)$.

Show that $a(t) = (1+i)^t$, where $i=a(1)-a(0)=a(1)-1.$

I concluded that from $a(s+t)=a(s)a(t), $ it must be an exponential function. But I can't figure out where $(1+i)$ came from for the equation $a(t) = (1+i)^t$. I understand that $i = a(1)-a(0)$ is a slope from time $t=0$ to $t=1$, but I still don't know how this equation is derived.

Answer 1

You have found the solution $a(t) = b^t$ for any $b>0$. Let $i = b-1$ and you recover the solution $(1+i)^t$. It is a convention to use $(1+i)$ as the base, to highlight the importance of the interest rate $i$.