高考数学之数列必背技巧

\[ \frac{1}{a_{k}a_{k+1}}=\frac{1}{d}\left(\frac{1}{a_{k}}-\frac{1}{a_{k+1}} \right) \] \[ \frac{1}{a_{k}a_{k+m}}=\frac{1}{md}\left(\frac{1}{a_{k}}-\frac{1}{a_{k+m}} \right) \] \[ \frac{1}{\sqrt{a_{k}}+\sqrt{a_{k+1}}} =\frac{1}{d}(\sqrt{a_{k+1}}-\sqrt{a_{k}}) \] \[ \frac{1}{\sqrt{a_{k}}+\sqrt{a_{k+m}}} =\frac{1}{md}(\sqrt{a_{k+m}}-\sqrt{a_{k}}) \] \[ \frac{n+1}{n^{2}(n+2)^{2}} =\frac{1}{4}\left[ \frac{1}{n^{2}}-\frac{1}{(n+2)^{2}} \right] \] \[ \frac{2^{n}}{(2^{n}-1)(2^{n+1}-1)} =\left[ \frac{1}{(2^{n}-1)}-\frac{1}{(2^{n+1}-1)} \right] \] \[ \frac{n2^{n}}{(n+1)(n+2)} =\left[ \frac{2^{n+1}}{(n+2)}-\frac{2^{n}}{n+1)} \right] \] \]