I have to use the principle of mathematical induction to show that $6^n - 5n+4$ is divisible by 5 for all integers $n>0$. No problem. I finished this part of the question.
The next thing I am asked to do is to make a formula that connects $b_n = 6^n - 5n+4$ and $b_{n+1}$. The problem isn't getting this done, but understanding what I am supposed to do since I can't figure out what they are looking after
So first step to deal with what you're asking for is to express $b_{n+1}$, where you get $$b_{n+1} = 6^{n+1} - 5n -1$$ Then you notice that there is an exponent of base $6$ in missing between $b_n$ and $b_{n+1}$, so try $$6b_n = 6^{n+1} - 30n + 24$$Then you deal with terms which are linear in $n$ (which is the $30n$ term appearing above). To get to be equal to $5n $ you do $$6b_n + 25n = 6^{n+1} - 5n + 24$$ Now the last thing to do is to get the constant terms equal to each other, by substracting $25$, namely $$6b_n + 25n -25 = 6^{n+1} - 5n - 1 = b_{n+1}$$
