Prove a sum of fractions less than a valueI happened to see this problem from an elementary school textbook, but cannot solve it: C = 1 5 + 1 6 + 1 7 + . . . + 1 15 + 1 16 + 1 17 Prove C &lt; 2Very much appreciate someone to enlighten me, especially with an elementary solution.

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Prove a sum of fractions less than a valueI happened to see this problem from an elementary school textbook, but cannot solve it:  C  =      1    5    +      1    6    +      1    7    +  .  .  .  +      1    15    +      1    16    +      1    17  Prove  C  &amp;lt;  2Very much appreciate someone to enlighten me, especially with an elementary solution.

vrtuljakc6 · Accepted Answer

they are usually comparing denominators to powers of two;sum(5,6,7)   &amp;lt;  3      /    4sum(8,9,10,11,12,13,14,15)   &amp;lt;  8      /    8  =  1sum(16,17)   &amp;lt;  2      /    16  =  1      /    8  C  &amp;lt;      15    8    &amp;lt;  2In the other direction:sum(5,6,7,8)   &amp;gt;  4      /    8  =  1      /    2sum(9,10,11,12,13,14,15,16)   &amp;gt;  8      /    16  =  1      /    2  C  &amp;gt;  1

EnvivyEvoxys6 · Accepted Answer

C  =      1    5    +      1    6    +      1    7    +  .  .  .  +      1    15    +      1    16    +      1    17    C  =      1    5    +      1    6    +      1    7    +      1    8    +      1    9    +      1    10    +      1    11    +      1    12    +      1    13    +      1    14    +      1    15    +      1    16    +      1    17    ≤      (          1      5        ×    6    )    +      (          1      10        ×    7    )    =      19    10    &amp;lt;  2

Prove a sum of fractions less than a value I happened to see this problem from an elementary school

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