Is this equation consistent with index notation? Is the equation $$a_i+b_j=c_k$$ consistent with index notation?
I think that the answer is yes, since the free index within each term is arbitrary, so it doesn't matter which one we pick. Is my thinking right, or should this be written as $a_i+b_i=c_i$?
 A: In addition to the previous answer, there are two kinds of indexes in tensor equations:  Free and Dummy.  Although you can change the identifier for a given index, free or dummy, you need to be careful in how you do this to maintain consistency.  For example, consider the following well known contravariant vector transformation equation:
$$
\bar{x}^i = \frac{\partial{\bar{x}^i}}{\partial{x}^j}x^j
$$
Where the index $i$ is the free index and the index $j$ is a dummy summation index.  If you change $i$ to something else like $r$ then you must do it on both sides of the equation so that the free indices balance on left and right side of the equation, such as:
$$
\bar{x}^r = \frac{\partial{\bar{x}^r}}{\partial{x}^j}x^j
$$
The dummy index $j$ can also be changed, such as to $s$ but the only occurrences that are affected are in the single term resulting in:
$$
\bar{x}^r = \frac{\partial{\bar{x}^r}}{\partial{x}^s}x^s
$$
Note that changing index identifiers is not arbitrary but usually done when combining terms such as in multiplying tensors that may involve summations or have in common the free index.
A: It needs to be written with all the indices the same
$$a_i+b_i=c_i$$
This is so that when you have more than one index you know how to match them up. For example
$$A_{ij}+B_{ij}$$
is different from
$$A_{ij}+B_{ji}$$
where I've swapped $i$ and $j$ on the $B$. To work out the first one you would just add the matrices, whereas to calculate the second one you would transpose $B$ and then add them.
