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

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

Let $n \in \Z$.

Then:

$\left \lfloor{\dfrac {n + 2 - \left \lfloor{n / 25}\right \rfloor} 3}\right \rfloor = \left \lfloor{\dfrac {8 n + 24} {25} }\right \rfloor$


Proof

\(\displaystyle \left \lfloor{\dfrac {n + 2 - \left \lfloor{n / 25}\right \rfloor} 3}\right \rfloor\) \(=\) \(\displaystyle \left \lfloor{\dfrac {\left({n - \left \lfloor{n / 25}\right \rfloor}\right) + 3 - 1} 3}\right \rfloor\) rearrangement
\(\displaystyle \) \(=\) \(\displaystyle \left \lceil{\dfrac {n - \left \lfloor{n / 25}\right \rfloor} 3}\right \rceil\) Floor of $\dfrac {m + n - 1} n$
\(\displaystyle \) \(=\) \(\displaystyle \left \lceil{\dfrac {n + \left \lceil{- n / 25}\right \rceil} 3}\right \rceil\) Ceiling of Negative equals Negative of Floor
\(\displaystyle \) \(=\) \(\displaystyle \left \lceil{\dfrac {\left \lceil{n - n / 25}\right \rceil} 3}\right \rceil\) Ceiling of Number plus Integer
\(\displaystyle \) \(=\) \(\displaystyle \left \lceil{\dfrac {\left \lceil{24 n / 25}\right \rceil} 3}\right \rceil\) simplification
\(\displaystyle \) \(=\) \(\displaystyle \left \lceil{\dfrac {8 n} {25} }\right \rceil\) Ceiling of $\dfrac {x + m} n$: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \left \lfloor{\dfrac {8 n + 25 - 1} {25} }\right \rfloor\) Floor of $\dfrac {m + n - 1} n$
\(\displaystyle \) \(=\) \(\displaystyle \left \lfloor{\dfrac {8 n + 24} {25} }\right \rfloor\) simplification

$\blacksquare$


Sources