$$ f(t)=\sum_{t_0=0}^{t} ab^{t_0} $$
In which $a$ and $b$ are constants, and $t$ is variable.
The purpose of the function is to describe the performance of a fixed income investment with monthly contributions. $a$ is the value that is added each month, $b$ is the monthly fee summed with 1, $t$ is the period in months, and $f(t)$ is the value of the investment after $t$ months.
$$ f(t)=kb^t+\sum_{t_0=0}^{t} ab^{t_0} $$
This is the same function considering there was an ammount already invested when $t=0$, represented by $k$.
These functions are not practical to be calculated without the help of a computer program, so I was wondering if there is a way of removing the summation from them.
As per the comments, the summation portion of the function can be expressed as the sum of a geometric progression:
$f(t)=kb^t+a(b^{t+1}-1)/(b-1);t≥0; b≠1$