Summation over k of Ceiling of k over 2
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Theorem
- $\ds \sum_{k \mathop = 1}^n \ceiling {\dfrac k 2} = \ceiling {\dfrac {n \paren {n + 2} } 4}$
Proof
By Permutation of Indices of Summation:
- $\ds \sum_{k \mathop = 1}^n \ceiling {\dfrac k 2} = \sum_{k \mathop = 1}^n \ceiling {\dfrac {n + 1 - k} 2}$
and so:
- $\ds \sum_{k \mathop = 1}^n \ceiling {\dfrac k 2} = \dfrac 1 2 \sum_{k \mathop = 1}^n \paren {\ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2} }$
First take the case where $n$ is even.
For $k$ odd:
\(\ds \ceiling {\dfrac k 2}\) | \(=\) | \(\ds \dfrac k 2 + \dfrac 1 2\) | Definition of Ceiling Function |
and:
\(\ds \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac {n + 1 - k} 2\) | Definition of Ceiling Function |
Hence:
\(\ds \ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac k 2 + \dfrac 1 2 + \dfrac {n + 1 - k} 2\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {k + 1 + n + 1 - k} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n + 2} 2\) |
For $k$ even:
\(\ds \ceiling {\dfrac k 2}\) | \(=\) | \(\ds \dfrac k 2\) | Definition of Ceiling Function |
and:
\(\ds \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac {n + 1 - k} 2 + \dfrac 1 2\) | Definition of Ceiling Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n - k + 2} 2\) |
Hence:
\(\ds \ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac k 2 + \dfrac {n - k + 2} 2\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {k + n - k + 2} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n + 2} 2\) |
So:
\(\ds \sum_{k \mathop = 1}^n \ceiling {\dfrac k 2}\) | \(=\) | \(\ds \dfrac 1 2 \sum_{k \mathop = 1}^n \paren {\ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \sum_{k \mathop = 1}^n \paren {\dfrac {n + 2} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 n \paren {\dfrac {n + 2} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n \paren {n + 2} } 4\) | Definition of Ceiling Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \ceiling {\dfrac {n \paren {n + 2} } 4}\) | as $\dfrac {n \paren {n + 2} } 4$ is an integer |
$\Box$
Next take the case where $n$ is odd.
For $k$ odd:
\(\ds \ceiling {\dfrac k 2}\) | \(=\) | \(\ds \dfrac k 2 + \dfrac 1 2\) | Definition of Ceiling Function |
and:
\(\ds \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac {n + 1 - k} 2 + \dfrac 1 2\) | Definition of Ceiling Function |
Hence:
\(\ds \ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac k 2 + \dfrac 1 2 + \dfrac {n + 1 - k} 2 + \dfrac 1 2\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {k + 1 + n + 1 - k + 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n + 3} 2\) |
For $k$ even:
- $\ceiling {\dfrac k 2} = \dfrac k 2$
and:
- $\ceiling {\dfrac {n + 1 - k} 2} = \dfrac {n + 1 - k} 2$
Hence:
\(\ds \ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2}\) | \(=\) | \(\ds \dfrac k 2 + \dfrac {n - k + 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {k + n - k + 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n + 1} 2\) |
Let $n = 2 t + 1$.
Then:
\(\ds \sum_{k \mathop = 1}^n \ceiling {\dfrac k 2}\) | \(=\) | \(\ds \dfrac 1 2 \sum_{k \mathop = 1}^n \paren {\ceiling {\dfrac k 2} + \ceiling {\dfrac {n + 1 - k} 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \sum_{k \mathop = 1}^{2 t + 1} \paren {\ceiling {\dfrac k 2} + \ceiling {\dfrac {2 t + 2 - k} 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac t 2 \dfrac {\paren {2 t + 1} + 1} 2 + \dfrac {t + 1} 2 \dfrac {\paren {2 t + 1} + 3} 2\) | there are $t$ even terms and $t + 1$ odd terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 t^2 + 2 t} 4 + \dfrac {2 t^2 + 6 t + 4} 4\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 t^2 + 8 t + 3} 4 + \dfrac 1 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2 t + 1} \paren {2 t + 3} } 4 + \dfrac 1 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n \paren {n + 2} } 4 + \dfrac 1 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ceiling {\dfrac {n \paren {n + 2} } 4}\) | Definition of Ceiling Function |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $36$