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# Harmonic Triangle

Take a look at the second diagonal:

$\frac{1}{2}$, $\frac{1}{6}$, $\frac{1}{12}$, $\frac{1}{20}$...

**Can you see a pattern?**

What fraction will appear in the second position on the nth row?

**Can you prove it?**

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Age 14 to 16

Challenge Level

*Harmonic Triangle printable worksheet*

**This is the start of the harmonic triangle:**

\begin{array}{ccccccccccc} & & & & &\frac{1}{1} & & & & & \\ & & & & \frac{1}{2} & & \frac{1}{2} & & & & \\ & & & \frac{1}{3} & &\frac{1}{6} & & \frac{1}{3} & & & \\ & & \frac{1}{4} & &\frac{1}{12} & & \frac{1}{12} & & \frac{1}{4} & & \\
& \frac{1}{5} & & \frac{1}{20} & & \frac{1}{30} & & \frac{1}{20} & & \frac{1}{5} & \\ \frac{1}{6} & & \frac{1}{30} & & \frac{1}{60} & & \frac{1}{60} & & \frac{1}{30} & & \frac{1}{6}\\ & & & & & \ldots& & & & & \end{array}

**Each fraction is equal to the sum of the two fractions below it.**

Look at the triangle above and check that the rule really does work.

Can you work out the next two rows?

The $n$th row starts with the fraction $\frac{1}{n}$, so the first diagonal goes:

$\frac{1}{1}$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$...

Take a look at the second diagonal:

$\frac{1}{2}$, $\frac{1}{6}$, $\frac{1}{12}$, $\frac{1}{20}$...

What fraction will appear in the second position on the nth row?

What about the third and fourth diagonals?

Click here for a poster of this problem.