Fractions differences in algebra

Context:It would normally follow on from work on sequences and fractions.

Question: Ruth was investigating fraction differences. She wrote down this sequence of fractions: 1/1, 1/2, 1/3, 1/4, 1,5 1,6 .. .. Then she worked out the difference between the consecutive fractions: 1/2, 1/6, 1/12, 1/20, 1/30, .. .. She Then worked out the differences between the Fractions in her second series. 1/3, 1/12 ... ... Investigate Further:

When Doing this make sure your look at these three things, 1:Making and monitoring decions to solve problems. 2.Communicating Mathematically. 3.Developing skills of Mathematical reasoning.

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...bers in the sequence
1/2, 1/6, 1/12, 1/20, 1/30, 1/42, 1/56,...
by adding in the denominator, without using the more difficult method of subtracting the elements from the first sequence.

The next sequence,
1/3, 1/12, ...
is too short to draw conclusions. I might guess that, since the difference between the denominators is 12-3=9, the next denominator might be 12+(9+4) or something like that, but I will have to make the sequence bigger by at least one or two elements to see the pattern. Getting more in the sequence by subtracting the elements in the second sequence gives more numbers:
1/3, 1/12, ...