I've noticed something about students versus more mathematically mature people when using induction. When we assume a statement is true for some $k$, i.e. $P(k)$ is true, we then proceed directly from $P(k)$ to $P(k+1)$. Whereas I've noticed in books that writers tend to do it differently; they tend to use the $P(k)$ statement somewhere in the middle, as opposed to starting with it. I'll make this clearer with an example:
I want to prove $$1+2+\ldots + n = n(n+1)/2$$
for all naturals $n$. Base case is obvious, suppose it's true for some $k$. Now from here, I would write down $1 + \ldots + k = k(k+1)/2$, then add $k+1$ to both sides, and derive $P(k+1)$ from there. However, I feel like I've noticed more advanced mathematicians start with $1 + \ldots + (k+1) = (1+\ldots + k) + (k+1)$, then invoke the $P(k)$ statement to get $(1 + \ldots + k) + (k+1) = k(k+1)/2 + (k+1)$, and then proceed from there. Is there anything wrong with my approach? Is there a reason mathematicians prefer the second way, or is that not true?