Recursive Form of Generalized Termial

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 $7$