Question

( \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + ......... + \(\dfrac{1}{19}\) + \(\dfrac{1}{20}\) ) . x = \(\dfrac{1}{19}\) + \(\dfrac{2}{18}\) + ........ + \(\dfrac{18}{2}\) + \(\dfrac{19}{1}\) 

Tìm x , lưu ý : dấu "." là dấu nhân

Answer 1

Lời giải:

$\frac{1}{19}+\frac{2}{18}+...+\frac{18}{2}+\frac{19}{1}$
$=(\frac{1}{19}+1)+(\frac{2}{18}+1)+....+(\frac{18}{2}+1)+(\frac{19}{1}+1)-19$

$=\frac{20}{19}+\frac{20}{18}+....+\frac{20}{2}+\frac{20}{1}-19$

$=20(\frac{1}{19}+\frac{1}{18}+.....+\frac{1}{2})+1$

$=20(\frac{1}{20}+\frac{1}{19}+\frac{1}{18}+...+\frac{1}{2})$

Ta có:

$(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{19}+\frac{1}{20}).x=20(\frac{1}{20}+\frac{1}{19}+\frac{1}{18}+...+\frac{1}{2})$

$\Rightarrow x=20$

Answer 2

\(\dfrac{1}{19}+\dfrac{2}{18}+...+\dfrac{18}{2}+\dfrac{19}{1}\)

\(=\left(\dfrac{1}{19}+1\right)+\left(\dfrac{2}{18}+1\right)+...+\left(\dfrac{18}{2}+1\right)+1\)

\(=\dfrac{20}{2}+\dfrac{20}{3}+...+\dfrac{20}{19}+\dfrac{20}{20}\)

\(=20\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{19}+\dfrac{1}{20}\right)\)

\(\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{19}+\dfrac{1}{20}\right)\cdot x=\dfrac{1}{19}+\dfrac{2}{18}+...+\dfrac{18}{2}+\dfrac{19}{1}\)

=>\(x\cdot\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{19}+\dfrac{1}{20}\right)=20\cdot\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{19}+\dfrac{1}{20}\right)\)

=>x=20