Recursive Form of Generalized Termial
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Theorem
The termial function as defined on the real numbers fulfils the identity:
- $x? = x + \paren {x - 1}?$
Proof
By definition of the termial on the real numbers:
- $x? = \dfrac {x \paren {x + 1} } 2$
Thus:
\(\ds x? - x\) | \(=\) | \(\ds \dfrac {x \paren {x + 1} } 2 - x\) | Definition of Termial | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x \paren {x + 1} - 2 x} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x^2 + x - 2 x} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x^2 - x} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {x - 1} x} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x - 1}?\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $7$