a) A=1/30+1/42+1/56+1/72+1/90+1/110+1/130

b)B= 3+3/2+3/2^2+....+3/2^9

`@answer`

`a) A=1/30+1/42+1/56+1/72+1/90+1/110+1/130`

$=\dfrac{1}{30}+\dfrac{1}{42}+\cdots+\dfrac{1}{132}\\ =\dfrac{1}{5.6}+\dfrac{1}{6.7}+\cdots+\dfrac{1}{11.12}\\ =\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\cdots+\dfrac{1}{11}-\dfrac{1}{12}\\ =\dfrac{1}{5}-\dfrac{1}{12}\\ =\dfrac{7}{60}$

`A = 3 +`$\frac{3}{2 }$ + $\frac{3}{2^{2} }$ +$\frac{3}{2^{3} }$ +...+$\frac{3}{2^{9} }$

`2A = 6+ 3 +` $\frac{3}{2 }$ + $\frac{3}{2^{2} }$ +$\frac{3}{2^{3} }$ +...+$\frac{3}{2^{8} }$

`2A - A = 6 -`$\frac{3}{2^{9} }$

`A= 6 -` $\frac{3}{2^{9} }$

$\text{a) A = }$$\dfrac{1}{30}$ + $\dfrac{1}{42}$ + $\dfrac{1}{56}$ + $\dfrac{1}{72}$ + $\dfrac{1}{90}$ + $\dfrac{1}{110}$ + $\dfrac{1}{130}$

    $\text{A = }$$\dfrac{1}{5 \times 6}$ + $\dfrac{1}{6 \times 7}$ + $\dfrac{1}{7 \times 8}$ + $\dfrac{1}{8 \times 9}$ + $\dfrac{1}{9 \times 10}$ + $\dfrac{1}{10 \times 11}$ + $\dfrac{1}{11 \times 12}$

     $\text{A = }$$\dfrac{1}{5}$ - $\dfrac{1}{6}$ + $\dfrac{1}{6}$ - $\dfrac{1}{7}$ + $\dfrac{1}{7}$ - $\dfrac{1}{8}$ + $\dfrac{1}{8}$ - $\dfrac{1}{9}$ + $\dfrac{1}{9}$ - $\dfrac{1}{10}$ + $\dfrac{1}{10}$ - $\dfrac{1}{11}$ + $\dfrac{1}{11}$ - $\dfrac{1}{12}$

     $\text{A = }$$\dfrac{1}{5}$ - ( $\dfrac{1}{6}$ + $\dfrac{1}{6}$ ) - ( $\dfrac{1}{7}$ + $\dfrac{1}{7}$ ) - ( $\dfrac{1}{8}$ + $\dfrac{1}{8}$ ) - ( $\dfrac{1}{9}$ + $\dfrac{1}{9}$ ) - ( $\dfrac{1}{10}$ + $\dfrac{1}{10}$ ) - ( $\dfrac{1}{11}$ + $\dfrac{1}{11}$ ) - $\dfrac{1}{12}$

     $\text{A = }$$\dfrac{1}{5}$ $\text{- 0 - 0 - 0 - 0 - 0 - 0 -}$ $\dfrac{1}{12}$

     $\text{A = }$$\dfrac{1}{5}$ `-` $\dfrac{1}{12}$ 

     $\text{A = }$$\dfrac{12}{60}$ `-` $\dfrac{5}{60}$ 

     $\text{A = }$$\dfrac{7}{60}$

Vậy $\text{A = }$$\dfrac{7}{60}$

b) `3+3/2+3/2^2+....+3/2^9`
$\Leftrightarrow$ $\dfrac{S}{3}$ = `1+1/2+1/2^2+....+1/2^9`

$\Leftrightarrow$ $\dfrac{S}{3}$ = `1+1/2+1/2^2+....+1/2^9`

$\Leftrightarrow$ $\dfrac{S}{3}$ = $\dfrac{1 - \dfrac{1}{2^10}}{1 - \dfrac{1}{2}}$

$\Leftrightarrow$ $\dfrac{S}{3}$ = $\text{2.( 1 -}$ $\dfrac{1}{2^10}$ )

$\Leftrightarrow$ $\text{S = 6 -}$ $\dfrac{3}{2^9}$

$\Leftrightarrow$ $\text{S = }$ $\dfrac{3069}{512}$

#YN_Yuri

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