Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions

In the setting of Euclidean Space and Lebesgue Measure: Standing Notation, let $q,N\in\mathbb{N}$. For $k\in[N]$ and $i\in[q]$ let $b(k,i)=(k-1)q+i$, an integer.

1. (Range)¶ $b(k,i)\in[qN]$ for every $k\in[N]$ and every $i\in[q]$.

2. (Bijection)¶ For every $j\in[qN]$ there is exactly one pair $(k,i)$ with $k\in[N]$, $i\in[q]$ and $b(k,i)=j$.

3. (Sums)¶ For every family $(a_{j})_{j\in[qN]}$ of real numbers,