# Element of Leibniz Harmonic Triangle is Sum of Numbers Below

## Theorem

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

## 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.

$\blacksquare$