A textbook uses induction to prove the binomial theorem and uses the following substitution:
$$\sum_{k=0}^{n-1} {n-1 \choose k} x^{k+1}y^{n-1-k} + \sum_{k=0}^{n-1} {n-1 \choose k} x^ky^{n-k}$$
Letting $i = k + 1$ in the first sum and $i = k$ in the second sum we find that
$$(x + y)^n = \sum_{i = 1}^{n} {n-1 \choose i-1} x^iy^{n-i} + \sum_{i = 0}^{n-1} {n-1 \choose i}x^iy^{n-i}$$
These terms are later joined into one summation and I am struggling to understand how we can later use the substituted i's interchangeably though they obviously replaced two different terms.
Note that the starting term in the first sum changed from $k=0$ to $i=1$, so the sums remained essentially the same. Similarly, $k$ in the first sum is replaced with $i-1$, whereas in the second sum it's replaced with $i$. However, the fact that we're evaluating the first sum starting at $i=1$ $($and $k=i-1 = 0)$ and finishing at $n$ (as opposed to $n-1$) means that the overall stays the same.