How to add function to a recursive equation?

Question

If I have the equations $f(x)=30c^x$ and $g(x)=g(x-1)+f(x)$, $g(0)=30$ why is $g(x)=30xc^x+30$ not the answer when something like $h(x)=h(x-1)+c$, $h(t)=d$ would make $h(x)=c(x-t)+d$? What I try to do goes as follows $30c^x(x-0)+30$ or $30xc^x+30$ but it does not have the same line or even shape. Here is my graphs desmos.com red=expected, purple=$f(x)$, green=unexpected. What am I doing wrong or is this something that is mathematically incorrect to try and do?

Answer 1

Simply because $h(x) = d $ is a constant function, you have predetermined the entire future and history of $h(x)$ in $x$. Whereas, $g(0)=30$ only says that at $x=0$, $g(0)=30$, but it's past and future are decided by $g(x) = g(x-1)+f(x)$. A simpler example could be $h_1(x) = 4$ which is a constant function, whereas h_2(0)=4 and h_2(x+1)=h_2(x)+4, $h_2(x)$ is a linearly growing function.