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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Sandy
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In the first set, you cancel both the 1/3, the 1/4, and the 1/5. You see the pattern now? This cancellation will continue until the last two terms (1/21 and 1/20 cannot be cancelled because n=20 is the limit).

In the first set you are left with 1/1 and 1/2. By the second set there's only 1/21 and 1/22. You add the first set and subtract the second. (1/1 + 1/2) - (1/21 + 1/22).