Đáp án:
$x=2008 $
Giải thích các bước giải:
$\dfrac{x}{2008}-\dfrac{1}{10}-\dfrac{1}{15}-\dfrac{1}{21}-..-\dfrac{1}{120}=\dfrac{5}{8} $
$⇒ \dfrac{x}{2008}-(\dfrac{1}{10}-\dfrac{1}{15}-\dfrac{1}{21}-..-\dfrac{1}{120})=\dfrac{5}{8} $
Xét: $\dfrac{1}{10}-\dfrac{1}{15}-\dfrac{1}{21}-..-\dfrac{1}{120} $
$=\dfrac{2}{20}-\dfrac{2}{30}-...-\dfrac{2}{240} $
$=2(\dfrac{1}{20}-\dfrac{1}{30}-...-\dfrac{1}{240}) $
$=2(\dfrac{1}{4×5}-\dfrac{1}{5×6}-...-\dfrac{1}{15×16} $
$ADCT:\dfrac{k}{n(n+k)}=\dfrac{1}{n}-\dfrac{1}{n+k} $
$=2(\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}+...+\dfrac{1}{15}-\dfrac{1}{16} $
$=2(\dfrac{1}{4}-\dfrac{1}{16}) $
$=2×\dfrac{3}{16} $
$=\dfrac{3}{8} $
Thay: $\dfrac{x}{2008}-\dfrac{3}{8}=\dfrac{5}{8} $
$⇒\dfrac{x}{2008}=\dfrac{3}{8}+\dfrac{5}{8} $
$⇒\dfrac{x}{2008}=1 $
$⇒x=2008 $
Vậy $x=2008 $
$#Nocopy$
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