Integer intervals with a summation

Question

I'm struggling to understand this question that was posed in a lecture. What is the value of the following sum of cardinalities? $\sum_{n=0}^{10} |[5i..(5i+3)]|$

I originally thought I would do the summation of 5i, which is 275 and then the summation of 5i+3 which is 308. I subtract the these and add 1, and get 34. But the answer is 44. And it has 4 unique elements.

Would appreciate an explanation for the process for obtaining this answer. Thank you

Answer 1

Assuming that $$[5i, 5i + 3] = \{5i, 5i+1, 5i +2, 5i+3\}$$ for $i \in [0,10]$:

Note that the above given set representation for $[5i, 5i+3]$ has 4 elements for each $i\in [0,10]$. Also, there are $11$ possible values of $i$.

So, the answer is $44$. Note that it's quite easy to check that there are no common elements between $[5i, 5i +3]$ and $[5j, 5j+3]$ for $i \neq j$.