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'Harmonic Triangle' printed from https://nrich.maths.org/
$$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$$
$$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$$
$$\frac{1}{12} = \frac{1}{20} + \frac{1}{30}$$
Look at the diagonal lines running from the right down to the left.
The fractions in the first one are $\frac{1}{1}, \frac{1}{2},
\frac{1}{3}, \frac{1}{4}$ and so on.
If $\frac{1}{n}$ is at the end of the nth row, the fraction above
it must be $\frac{1}{n-1}$ and the fraction below it must be
$\frac{1}{n+1}$.
Have a look at the second diagonal (the one formed by taking the
second number in each row: it starts $\frac{1}{2}, \frac{1}{6},
\frac{1}{12}, \frac{1}{20}$.
Can you find a pattern for these numbers so that you can work them
out easily (without having to subtract fractions)?
Can you explain why the pattern works?