Dumb question: what is $\sum\limits_{n = 1}^k 1$ Let $\{I_n\}$ be a collection of intervals on $\Bbb {R}$, whose length denoted by $I_n = |I_n|$
Then what is $\sum\limits_{n = 1}^k (I_n + \alpha)$, where $\alpha$ is some real number?
The notation confuses me:
$$\sum_{n = 1}^k (I_n + \alpha) = \sum_{n = 1}^k I_n +  \sum_{n = 1}^k \alpha = \sum_{n = 1}^k I_n +  \alpha \sum_{n = 1}^k 1$$
What is $\sum\limits_{n = 1}^k 1$?
Confusion 1: $n$ was an index, there is no index in the $\sum\limits_{n = 1}^k 1$ term...
Confusion 2:$\sum\limits_{n = 1}^k 1 = 1 + 1 +\cdots + 1 = k$? or does that sum term does nothing since it is not summing over something indexed by $n$?$

If you translate an interval by $\alpha$, it's length stays the same. Suppose $I=[3,4]$ and $\alpha=5$, $I+\alpha=[8,9]\implies |I+\alpha|=9-8=1=|I|$. Note that $|I+\alpha|\neq |I|+\alpha$.
So
$$
\sum_{n=1}^k |I_n+\alpha|=\sum_{n=1}^k|I_n|=\sum_{n=1}^k|I_1|=|I_1|\sum_{n=1}^k1=k|I
_1|
$$
