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Giả sử $d$ là công sai của $(u_n)$.
Ta có:
$\frac{1}{u_{1}.u_{2}} +\frac{1}{u_{2}.u_{3}}+...+\frac{1}{u_{n-1}.u_{n}}$
$=\frac{1}{d}(\frac{u_2-u_1}{u_{1}.u_{2}} +\frac{u_3-u_2}{u_{2}.u_{3}}+...+\frac{u_n-u_{n-1}}{u_{n-1}.u_{n}})$
$=\frac{1}{d}(\frac{1}{u_1}-\frac{1}{u_2}+\frac{1}{u_2}-\frac{1}{u_3}+...+\frac{1}{u_{n-1}}-\frac{1}{u_n})$
$=\frac{1}{d}(\frac{1}{u_1}-\frac{1}{u_n})$
$=\frac{1}{d}.\frac{u_1+(n-1)d-u_1}{u_n.u_1}=\frac{n-1}{u_n.u_1}$
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