Problem

\( \sum_{n=1}^{a-1} \frac{1}{2}\left(\frac{1}{n}-\frac{1}{n+2}\right) \)

Answer

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Answer

\(=\frac{1}{2}\left(\frac{1}{1}+\frac{1}{2}-\frac{1}{a} - \frac{1}{a + 1}\right) \)

Steps

Step 1 ：\(=\frac{1}{2}\sum_{n=1}^{a-1}\left(\frac{1}{n}-\frac{1}{n+2}\right) \)

Step 2 ：\(=\frac{1}{2}\left(\left(\frac{1}{1}-\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\cdots+\left(\frac{1}{a-1}-\frac{1}{a+1}\right)\right) \)

Step 3 ：\(=\frac{1}{2}\left(\frac{1}{1}+\frac{1}{2}-\frac{1}{a} - \frac{1}{a + 1}\right) \)

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