Correct denotation for sum function

Question

The variable $x(t)$ can assume the values 0 or 1 at every time-slot (minute) $t$. The sum of all $x$ over the course of one week has to be at least 25 (every week! meaning there can never be a 7 day period without at least 25).

I formulated this as follows: $$\sum_{t}^{t+7~days}x(t) \geq 25$$ Someone (no mathematician) indicated that the correct form for this might be: $$\sum\nolimits_{t=i}^{i+7~days}x(t) \geq 25$$

Is this correct? What is the difference between the two?

Answer 1

We could write \begin{align*} \sum_{t=i}^{i+7\cdot24\cdot 60-1}x(t)\geq 25\qquad i\in\mathbb{Z},\ i\geq 0 \end{align*}