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can all fractions with a numerator of two be written as the sum of two different unitary fractions with mathematical proof

 Jan 7, 2020
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egyptian fractions

can all fractions with a numerator of two be written as the sum of two different unitary fractions with mathematical proof


I assume:
\(\dfrac{2}{n}=\dfrac{1}{a}+\dfrac{1}{b} \)

 

if \(n\) is even then we set \(n=2k\):

\(\mathbf{\dfrac{2}{n}}=\dfrac{2}{2k}=\dfrac{1}{k}\mathbf{=\dfrac{1}{k+1} + \dfrac{1}{k(k+1)} }\)

 

if \(n\) is odd then we set \(n=2k-1\):

\(\mathbf{\dfrac{2}{n}}=\dfrac{2}{2k-1}\mathbf{=\dfrac{1}{k} + \dfrac{1}{k(2k-1)} }\)

 

 

laugh

 Jan 7, 2020

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