I have a couple of questions regarding the numbering of indices for vectors.
For a the vector $ \mathbf a=(a_1, a_2, \dots, a_n) $, I know this is a tuple of $n$ components.
But if I have $$ \mathbf b=(b_0, b_1, \dots, b_{n-1}) \tag 1 $$ is this a a tuple of $n-1$ components?
And $$ \mathbf c=(c_2, c_3, \dots, c_{n+1}) \tag 2 $$ is this a a tuple of $n-1$ components?
But if the first index is $1$ and last is $n+1$, i.e. $$ \mathbf d=(d_1, d_2, \dots, d_{n+1}) \tag 3 $$ is this a a tuple of $n+1$ components?
Or if the first index is $0$ and last is $n+1$, i.e. $$ \mathbf e=(e_0, e_1, \dots, e_{n+1}) \tag 4 $$ is this a a tuple of $n+2$ components?
And finally, if the first index is $2$ and last is $n$, i.e. $$ \mathbf f=(f_2, f_3, \dots, f_{n}) \tag 5 $$ is this a a tuple of $n-1$ components?
The number of components in $(b_f, b_{f+1}, \dots, b_l)$ is always $l-f+1$ (last $-$ first $+1$). So, (1), (2), (3), (4), (5) have $n,n,n+1,n+2, n-1$ components respectively. Always try things out for small $n$, and check your answers.