Floor of m+n-1 over n/Example 1

Example of use of Floor of $\frac {m + n - 1} n$

Let $n \in \Z$.

Then:

$\floor {\dfrac {n + 2 - \floor {n / 25} } 3} = \floor {\dfrac {8 n + 24} {25} }$

$\ds \floor {\dfrac {n + 2 - \floor {n / 25} } 3}$

$=$

$\ds \floor {\dfrac {\paren {n - \floor {n / 25} } + 3 - 1} 3}$

rearrangement

$\ds $

$=$

$\ds \ceiling {\dfrac {n - \floor {n / 25} } 3}$

$\ds $

$=$

$\ds \ceiling {\dfrac {n + \ceiling {-n / 25} } 3}$

$\ds $

$=$

$\ds \ceiling {\dfrac {\ceiling {n - n / 25} } 3}$

$\ds $

$=$

$\ds \ceiling {\dfrac {\ceiling {24 n / 25} } 3}$

simplification

$\ds $

$=$

$\ds \ceiling {\dfrac {8 n} {25} }$

$\ds $

$=$

$\ds \floor {\dfrac {8 n + 25 - 1} {25} }$

$\ds $

$=$

$\ds \floor {\dfrac {8 n + 24} {25} }$

simplification

$\blacksquare$