Cho A= 1/(1*2) + 1/2*3 +... +1/29*30 + 1/30*31 Và B = 1/1*4 + 2/4*10 + ...+ 3/10*19 + 4/19*31

* Hướng dẫn giải

A$=\frac{1}{1.2}+\frac{1}{2.3}+\mathrm{\dots }+\frac{1}{29.30}+\frac{1}{30.31}$

$=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\mathrm{\dots }+\frac{30-29}{29.30}+\frac{31-30}{30.31}$

$=(\frac{2}{1.2}-\frac{1}{1.2})+(\frac{3}{2.3}-\frac{2}{2.3})+$$\mathrm{\dots }+(\frac{30}{29.30}-\frac{29}{29.30})+(\frac{31}{30.31}-\frac{30}{30.31})$

$=(\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+\mathrm{\dots }+(\frac{1}{29}-\frac{1}{30})+(\frac{1}{30}-\frac{1}{31})$

$=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\mathrm{\dots }+\frac{1}{29}-\frac{1}{30}+\frac{1}{30}-\frac{1}{31}$

$=1+(-\frac{1}{2}+\frac{1}{2})+(-\frac{1}{3}+\frac{1}{3})+$$\mathrm{\dots }+(-\frac{1}{30}+\frac{1}{30})-\frac{1}{31}$

$=1+0+\mathrm{\dots }+0-\frac{1}{31}$

$=1-\frac{1}{31}$

$=\frac{31-1}{31}$

$=\frac{30}{31}$

B$=\frac{1}{1.4}+\frac{2}{4.10}+\frac{3}{10.19}+\frac{4}{19.31}$

$=\frac{1}{1}.\frac{1}{4}+\frac{1}{4}.\frac{2}{10}+\frac{1}{19}.\frac{3}{10}+\frac{1}{19}.\frac{4}{31}$

$=\frac{1}{4}(\frac{1}{1}+\frac{2}{10})+\frac{1}{19}(\frac{3}{10}+\frac{4}{31})$

$=\frac{1}{4}(1+\frac{1}{5})+\frac{1}{19}(\frac{3.31}{10.31}+\frac{4.10}{31.10})$

$=\frac{1}{4}(\frac{5}{5}+\frac{1}{5})+\frac{1}{19}(\frac{93}{310}+\frac{40}{310})$

$=\frac{1}{4}.\frac{6}{5}+\frac{1}{19}.\frac{133}{31}$

$=\frac{2.3}{\mathrm{2.2.5}}+\frac{19.7}{19.31}$

$=\frac{3}{10}+\frac{7}{31}$$=\frac{3.31}{10.31}+\frac{7.10}{10.31}$

$=\frac{93}{310}+\frac{70}{310}$

$=\frac{163}{310}$