Traditional mystery can be an odd genre. Series mystery more so.
There are no writing rules, but there are useful guidelines you can apply or ignore at will. iTools Not Rules.
One of these guidelines is that the protagonist should experience an “arc”: they start in some condition or belief, they get tested, and they emerge with that condition or belief changed. They might give it up. They might modify it. They might discover why it is. They might even strengthen it. Your protagonist doesn’t have to have an arc, but it’s a powerful technique. (Readers have observed that Nick Aames in The Last Dance doesn’t have much of an arc. But Nick is never the protagonist in his stories. He’s a force of nature who tests the protagonist.)
But in traditional mysteries where a detective finds clues and solves a mystery, the detective CAN have an arc, but frequently they don’t. The killer has an arc. The victim or their survivors have arcs. And the detective has a strong influence on these arcs (much like Nick). But unless the mystery is personal to the detective, the detective may not change.
Now “personal” can have a lot of interpretations. It might be a personal attachment to the victim or the accused or the killer. It might also be that the detective is challenged. Honor and reputation are at stake. It might not even be the A story (the mystery), but rather a B story about the detective’s life outside the case.
And this is even more true in series mysteries where the same detective appears in episode after episode. Does Columbo change in every episode? Not really. He’s challenged in every episode, and sometimes even stymied by a powerful killer; but in the end, he triumphs, and he’s essentially the same character he was at the end of the last episode. I can think of two times when I really saw a change: the winery episode where he came to like and even sympathize with the killer; and the episode where a ruthless foe plotted to kill his wife. I’m surely missing a few, but he was mostly unchanging.
In Bobo Buttons, Private Eye, Bobo may not have an arc in every book. I won’t know until I write them. In Route Book 1, his arc is learning that he’s a detective. In Route Book 2, his arc is learning to be in a functioning relationship. In Route Book 3, he’s discovering how important community is to him by being deprived of it. There are future arcs I can’t discuss without spoilers, but I’m not sure Bobo will have an arc in Route Book 4. He’ll be a confident detective with a challenging case, but maybe no arc.
Yet you can also have series arcs, changes that span multiple episodes. And that Bobo has, and I can say that without spoilers. He starts Route Book 1 with no intention of being a detective. By Route Book 3, he realizes he’s stuck with it. In 4 he’ll embrace it. In later Route Books, he’ll discover what it means when his world expects him to be a detective first and foremost. In Route Book 10, he’ll be confronted with a choice that reveals just how much he has changed. And in 11 and 12, he’ll be working at a new level—with more dangerous opponents.
You can see how it starts here!
a.co/d/04dbCL2