Can sums be manipulated in this way? I'm working on an induction proof and I'm trying to manipulate a sum so that I can use my inductive hypothesis. Is the following possible?
$$\sum_{i=1}^{3n+4} i = \sum_{i=1}^{3n+1+3}i = \sum_{i=1}^{3n+1}i + \sum_{i=3n+2}^{3n+4}i$$

Of course. Even more generally, using a generic summand, you're just saying
$$a_1+\cdots+a_{k+n}=(a_1+\cdots+a_k)+(a_{k+1}+\cdots+a_{k+n})$$
