We get \frac{1}{n} = \frac{a}{n(a-n)}-\frac{1}{a-n}
Putting a= 83 we get
\frac{1}{n} = \frac{83}{n(83-n)}-\frac{1}{83-n}
Running n from 29 upto 55 in increment of 2 we have
\sum_{n=14}^{27} \frac{1}{2n+1}=\sum_{n=14}^{27} (\frac{83}{(2n+1)(82-2n)}-\frac{1}{82-2n})
Or \sum_{n=14}^{27} \frac{1}{2n+1}=83\sum_{n=14}^{27}\frac{1}{(2n+1)(82-2n)}- \sum_{n=14}^{27}\frac{1}{82-2n}
As \sum_{n=14}^{27}\frac{1}{(2n+1)(82-2n)} when we take the sum we shall get it of the form \frac{p}{q} where q does not have a divisor 83. as 83 does not divide any denominator
So \sum_{n=14}^{27} \frac{1}{2n+1}=\frac{83p}{q}- \sum_{n=14}^{27}\frac{1}{82-2n}\cdots(1)
Further 82-2n = 2k => n = 41-k
So \sum_{n=14}^{27}\frac{1}{82-2n} = \sum_{n=14}^{27}\frac{1}{2n}\cdots(2)
Now the given expression
= \sum_{n=0}^{27} \frac{1}{2n+1 } - \sum_{n=1}^{27} \frac{1}{2n}
= \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{27} \frac{1}{2n} + \sum_{n=14}^{27} \frac{1}{2n+1 }
= \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{27} \frac{1}{2n} + \frac{83p}{q}- \sum_{n=14}^{27}\frac{1}{82-2n}
= \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{27} \frac{1}{2n} + \frac{83p}{q}- \sum_{n=14}^{27}\frac{1}{2n}
= \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{13} \frac{1}{2n} + \frac{83p}{q}- 2 \sum_{n=14}^{27}\frac{1}{2n}
= \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{13} \frac{1}{2n} + \frac{83p}{q}- \sum_{n=14}^{27}\frac{1}{n}
= \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{13} \frac{1}{2n} + \frac{83p}{q}- \sum_{n=7}^{13}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{2n+1}
= (\sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=7} ^{13} \frac{1}{2n+1}) + \frac{83p}{q}- \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=7}^{13}\frac{2}{2n}
= \sum_{n=0}^{6} \frac{1}{2n+1 } + \frac{83p}{q}- \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{n}
= \frac{83p}{q} + \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=1}^{13} \frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{2n+1}
= \frac{83p}{q} + \sum_{n=0}^{13} \frac{1}{2n+1 } - \sum_{n=7}^{13} \frac{1}{2n+1} - \sum_{n=1}^{13}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{2n}
= \frac{83p}{q} + \sum_{n=0}^{6} \frac{1}{2n+1 } - \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{2n}- \sum_{n=7}^{13}\frac{1}{2n}
= \frac{83p}{q} + \sum_{n=0}^{6} \frac{1}{2n+1 } - \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{n}
= \frac{83p}{q} + \sum_{n=0}^{6} \frac{1}{2n+1 } - \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{n}
= \frac{83p}{q} + \sum_{n=0}^{6} \frac{1}{2n+1 } - \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=7}^{13}\frac{1}{n}
= \frac{83p}{q} + \sum_{n=0}^{6} \frac{1}{2n+1 } - \sum_{n=1}^{6}\frac{1}{2n} - (\sum_{n=3}^{6}\frac{1}{2n+1} + \sum_{n=4}^{6}\frac{1}{2n})
= \frac{83p}{q} + \sum_{n=0}^{2} \frac{1}{2n+1 } - \sum_{n=1}^{6}\frac{1}{2n} - \sum_{n=4}^{6}\frac{1}{2n}
= \frac{83p}{q} + \sum_{n=0}^{2} \frac{1}{2n+1 } - \sum_{n=1}^{3}\frac{1}{2n} - \sum_{n=4}^{6}\frac{1}{2n} - \sum_{n=4}^{6}\frac{1}{2n}
= \frac{83p}{q} + \sum_{n=0}^{2} \frac{1}{2n+1 } - \sum_{n=1}^{3}\frac{1}{2n} - \sum_{n=4}^{6}\frac{1}{n}
= \frac{83p}{q} + 1 + \frac{1}{3} + \frac{1}{5} - \frac{1}{2} - \frac{1}{4} - \frac{1}{6} - \frac{1}{4} - \frac{1}{5} - \frac{1}{6}
=\frac{83p}{q}
Numerator is divisible by 83 and denominator is not and 83 is prime . hence proved
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