Translation of Index Variable of Summation/Examples

Examples of Translation of Index Variable of Summation

 * $\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j = \sum_{1 \mathop \le j - 1 \mathop \le n} a_{j - 1} = \sum_{2 \mathop \le j \mathop \le n + 1} a_{j - 1}$

Proof
A direct instance of Translation of Index Variable of Summation.