Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 60

Assume that you invest $x$ dollars at 4% interest compounded annually, then the amount $A(x)$ of the investment after one year is $A(x) = 1.04x$. Find $A \circ A$, $A \circ A \circ A$, and $A \circ A \circ A \circ A$. What do these compositions represent? Write a formula for the composition of $n$ copies of $A$.



$
\begin{equation}
\begin{aligned}


A \circ A =& 1.04 (1.04x) &&= 1.0816x , \quad \text{ total investment after 2 years.}\\

A \circ A \circ A =& 1.04 (1.04)(1.04x) &&= 1.249x, \quad \text{ total investment after 3 years.}\\

A \circ A \circ A \circ A =& 1.04(1.04)(1.04)(1.04x) &&= 1.699x , \quad \text{ total investment after 4 years.}\\

\end{aligned}
\end{equation}
$



These compositions represent the growth of interest after $"n"$ years. Therefore, the total investment after $"n"$ years can be written as $ \cdots$


$A(x) = 1.04^n (x)$