Suppose we have a set of numbers $T_{ij}=B_{ij}+a_i$ and another set $A_{ij}$.
Now, according to index notation, $A_{ij}T_{ij}=A_{ij}B_{ij}+A_{ij}a_i$.
The $1^{st}$ term in the RHS is a summation over both indices $i$ and $j$ while the $2^{nd}$ is a summation over $i$ only.
However, when calculating the value without index notation, we get $$A_{ij}T_{ij}=\sum_i\sum_jA_{ij}T_{ij}+A_{ij}a_i$$Both terms are summed up over $i$ and $j$. So where is the problem ?