Fraction Sum SeriesThis question was asked in (selection) IMO for 8th graders. 1 / 2 + 1 / 6 + 1 / 12 + 1 / 20 + 1 / 30 + 1 / 42 + 1 / 56 + 1 / 72 + 1 / 90 + 1 / 110 + 1 / 132I have noticed that it can be written as 1 / ( 1 ∗ 2 ) + 1 / ( 2 ∗ 3 ) + 1 / ( 3 ∗ 4 ) + 1 / ( 4 ∗ 5 ) . . . . + 1 / ( 11 ∗ 12 )However I don't know how to continue..

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Fraction Sum SeriesThis question was asked in (selection) IMO for 8th graders.  1      /    2  +  1      /    6  +  1      /    12  +  1      /    20  +  1      /    30  +  1      /    42  +  1      /    56  +  1      /    72  +  1      /    90  +  1      /    110  +  1      /    132I have noticed that it can be written as   1      /    (  1  ∗  2  )  +  1      /    (  2  ∗  3  )  +  1      /    (  3  ∗  4  )  +  1      /    (  4  ∗  5  )  .  .  .  .  +  1      /    (  11  ∗  12  )However I don&#039;t know how to continue..

Janiyah Patton · Accepted Answer

Hint:Your sum can be written in the following form:      ∑          n      =      1              11                  1              n        (        n        +        1        )            Apply partial fraction decomposition to       1          n      (      n      +      1      )       to write it in the form       A    n    +      B          n      +      1       and see what happens as you simplify.The series should telescope. I.e. adjacent terms will cancel with one another, leaving you with only the first and last terms not cancelled. You have       1          n      (      n      +      1      )        =      1    n    −      1          n      +      1      . The series then looks like       1    1    −      1    2    +      1    2    −      1    3    +      1    3    −      1    4    +      1    4    −  ⋯  −      1    11    +      1    11    −      1    12

dikcijom2k · Accepted Answer

Then note that       1          1      ⋅      2       =       1    1    −      1    2   and the sum telescopes.

Fraction Sum Series This question was asked in (selection) IMO for 8th graders. 1 <mrow cl

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