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Odd run sums and fractions

Published online by Cambridge University Press:  20 June 2025

Ron Knott*
Affiliation:
92 Pennine Road, Horwich, Bolton, BL6 7HW e-mail: ronknott@mac.com

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A series of consecutive odd numbers has interesting properties useful for classroom investigation [1]. This Article was inspired by the series of fractions for ${{1}\over{2}}$ using only sums of consecutive odd numbers

$$\frac{1}{3} = \frac{{1 + 3}}{{5 + 7}} = \frac{{1 + 3 + 5}}{{7 + 9 + 11}} = {\text{ }} \ldots {\text{ }} = \frac{{1 + 3 + {\text{ }} \ldots {\text{ }} + (2n - 3) + (2n - 1)}}{{(2n + 1) + (2n + 3) + {\text{ }} \ldots {\text{ }} + (2n + (2n - 1))}}.$$

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Articles
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© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association