Self referential functions

Question

I am currently attempting to work with a particularly annoying self-referential function; that is the function is defined in terms of itself and also takes $x$ as an input. This function is defined as follows: $$b_{k+1}(x) = 3 \cdot b_k(x) + f(b_k(x)),\ b_1(x)=3x+1$$ The definition of $f(x)$ is not important here, but note that $f(x)$ cannot be simplified in any way that matters. I want to show that $$b_k(x)=3^k x + 3^{k-1} + \sum_{n=2}^k 3^{k-n} \cdot f(b_{n-1}(x))$$ However, I do not know how I would come to this conclusion, or even if I have to prove it. I simply found it by observation. Looking around, I have found very little information about self-referential functions, only a lot about Tupper's self-referential formula.
How would I/should I prove this?