# Element of Leibniz Harmonic Triangle is Sum of Numbers Below

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## Theorem

The elements in the Leibniz harmonic triangle are the sum of the elements immediately below them.

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## Proof

By definition of Leibniz harmonic triangle, element $\tuple {n, m}$ is:

- $\dfrac 1 {\paren {n + 1} \binom n m}$

Thus we have:

\(\ds \) | \(\) | \(\ds \dfrac 1 {\paren {n + 2} \binom {n + 1} m} + \dfrac 1 {\paren {n + 2} \binom {n + 1} {m + 1} }\) | Elements of Leibniz Harmonic Triangle immediately below | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\binom {n + 1} {m + 1} + \binom {n + 1} m} {\paren {n + 2} \binom {n + 1} m \binom {n + 1} {m + 1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\binom {n + 2} {m + 1} } {\paren {n + 2} \binom {n + 1} m \binom {n + 1} {m + 1} }\) | Pascal's Rule | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\frac {\paren {n + 2}!} {\paren {m + 1}! \paren {n - m + 1}!} } {\paren {n + 2} \frac {\paren {n + 1}!} {m! \paren {n - m + 1}!} \frac {\paren {n + 1}!} {\paren {m + 1}! \paren {n - m}!} }\) | Definition of Binomial Coefficient | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\paren {n + 1}!} {\frac {\paren {n + 1}!} {m!} \frac {\paren {n + 1}!} {\paren {n - m}!} }\) | some simplification | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac 1 {\frac {\paren {n + 1}!} {m! \paren {n - m}!} }\) | further simplification | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac 1 {\paren {n + 1} \frac {n!} {m! \paren {n - m}!} }\) | further simplification | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac 1 {\paren {n + 1} \binom n m}\) | Definition of Binomial Coefficient |

Hence the result.

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $35$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $35$