### Do it more than once

Sometimes it is desirable to perform a computation several times. Consider a situation when you want to determine the future value of your savings, after a determined number of years into the future. Using some symbols and equations we can express this as an iterative computation as follows.

Assume that you invest an amount of *a* units
of money, e.g. dollars, today, and that it is desired to find
out how much this investment is worth after a number of
years. Let the number of years be expressed using a variable
*n*. If the *interest rate* is *p* percent, then
the amount available after one year is
$$
a_1 = a + a \cdot \frac{p}{100}
\quad \quad \quad (5)
$$
This equation can be rewritten, as
$$
a_1 = a \left (1 + \frac{p}{100} \right )
\quad \quad \quad (6)
$$
We see that, after two years, the amount becomes
$$
a_2 = a \left (1 + \frac{p}{100} \right )^2
\quad \quad \quad (7)
$$
and if we dare to generalize, we can figure out the amount after
*n* years, as
$$
a_n = a \left (1 + \frac{p}{100} \right )^n
\quad \quad \quad (8)
$$

Suppose now that the interest rate varies, as in the real life, from
year to year. We can use the notation \(p_i\) for the interest
rate during year number \(i\). Using a more
advanced notation, we can now express the amount after *n*
years as
$$
a_n = a \prod_{i = 1}^n \left (1 + \frac{p_i}{100} \right )
\quad \quad \quad (9)
$$