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Egyptian Fractions

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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Weekly Problem 44 - 2013

Weekly Problem 44 - 2013

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Weekly Problem 26 - 2008

If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

Harmonic Triangle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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 $\frac{1}{n}$.
 
We can continue the first diagonal ($\frac{1}{1}$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, and so on) using this rule.

Take a look at the second diagonal: ($\frac{1}{2}$, $\frac{1}{6}$, $\frac{1}{12}$, $\frac{1}{20}$, and so on).
What do you notice about the numerators and denominators of these fractions?

Can you prove the pattern will continue?

What about the third and fourth diagonals?

 

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