Here we have \(an= \frac{1}{n} - \frac{1}{(n+2)}\). The idea is to find a pattern in the sum
\(a1= \frac{1}{1} - \frac{1}{3}\)
\(a2= \frac{1}{2} - \frac{1}{4}\)
\(a3= \frac{1}{3} - \frac{1}{5}\)
\(a4= \frac{1}{4} - \frac{1}{6}\)
Now we have the first term from a3 can cancels second term from previous numbers
Now observing the last few terms...
\(a18= \frac{1}{18} - \frac{1}{20}\)
\(a19= \frac{1}{19} - \frac{1}{21}\)
\(a20= \frac{1}{20} - \frac{1}{22}\)
Now we have here the last two terms namely \(\frac{1}{21}\) and \(\frac{1}{22}\) are not cancelled.
\((1+\frac{1}{2}) - (\frac{1}{21}+\frac{1}{22})\)
Hence the sum is
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