Interesting pattern when you look at the sums:

First 2: \(\displaystyle \frac{1}{3}+\frac{1}{6} =\frac{1}{2}\)

First 3: \(\displaystyle \frac{1}{3}+\frac{1}{6} +\frac{1}{10}=\frac{3}{5}\)

First 4: \(\displaystyle \frac{2}{3}\)

First 5: \(\displaystyle \frac{5}{7}\)

First 6: \(\displaystyle \frac{3}{4}\)

First 7: \(\displaystyle \frac{7}{9}\)

Note the two "patterns" for sums of evens and odds.

Since 231 is the 20th number in the triangular sequence (starting with 3 ie, 3, 6, 10, ...), we are particularly interested in the sum of evens.

First 2: \(\displaystyle \frac{1}{2}\) ... numerator is half of 2, denominator is 1 more

First 4: \(\displaystyle \frac{2}{3}\) ... numerator is half of 4, denominator is 1 more

First 6: \(\displaystyle \frac{3}{4}\) ... numerator is half of 6, denominator is 1 more

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First 20: \(\displaystyle \frac{10}{11}\) ... numerator is half of 20, denominator is 1 more

Beautiful!!