I'm currently trying to find the result of this sum
$\sum_{i=1}^n \lfloor\sqrt{i}\rfloor$
using a limited subset of sum "properties" that you can see here
I used to do a lot of variable change in my calculus class and I'm trying to apply the same approach here using this video as a guide
With let z = $\lfloor\sqrt{i}\rfloor$ I obtain the following sum (which is great as I could use the second property to find the result)
$i = 1: z = \lfloor\sqrt{1}\rfloor = 1$
$i = n: z = \lfloor\sqrt{n}\rfloor$
$\sum_{z=1}^{\lfloor\sqrt{n}\rfloor} z$
Which I know is a mistake already because with n = 7 the first summation gives 7 and the second gives 3... It's really probably a dumb mistake but I'm unable to find it.