Saturday, January 3, 2015

362: Adding Fractions Again

How can we approach this beast of a problem without finding LCM?

$\dfrac{1}{1} - \dfrac{5}{6} + \dfrac{7}{12} - \dfrac{9}{20} + \dfrac{11}{30} - \dfrac{13}{42} + \dfrac{15}{56} - \dfrac{17}{72} + \dfrac{19}{90} = \dfrac{a}{b}$

1 comment:

  1. Rewrite 1/1 as -1/2 + 3/2. The sequence 3/2 - 5/6 + … + 19/90 is
    (-1)^(n+1)(2n + 1)/(n(n+1)), n = 1,2,…,9. Take differences pairwise, then add the results pairwise, then do it again to find the eight terms sum to 8/9 = 80/90. Doing the algebra and finding the final answer is left as an exercise for the mathematically literate.

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