5/2+5/6+5/18+5/54+5/162+5/486

`5/2 + 5/6 + 5/18 + 5/54 + 5/162 + 5/486`

= `( 5/2 + 5/6 ) + ( 5/18 + 5/54 ) + ( 5/162 + 5/486 )`

= `5 . ( 1/2 + 1/6 ) + 5 . ( 1/18 + 1/54 ) + 5 . ( 1/162 + 1/486 )`

= `5 . ( 1/1.2 + 1/2.3 ) + 5/3 . ( 1/3.6 + 1/6.9 ) + 5/9 . ( 1/9.18 + 1/18.27 )`

= `5 . ( 1 - 1/2 + 1/2 - 1/3 ) + 5/3 . ( 1/3 - 1/6 + 1/6 - 1/9 ) + 5/9 . ( 1/9 - 1/18 + 1/18 - 1/27 )`

= `5 . ( 1 - 1/3 ) + 5/3 . ( 1/3 - 1/9 ) + 5/9 . ( 1/9 - 1/27 )`

= `5 . 2/3 + 5/3 . 2/9 + 5/9 . 2/27`

= `10/3 + 10/27 + 10/243`

= `100/27 + 10/243`

= `910/243`

Đáp án:

 \(\dfrac{910}{243}\)

Giải thích các bước giải:

\[\begin{array}{l}
\frac{5}{2} + \frac{5}{6} + \frac{5}{{18}} + \frac{5}{{54}} + \frac{5}{{162}} + \frac{5}{{486}}\\
 = 5 \times \left( {\frac{1}{{1 \times 2}} + \frac{1}{{2 \times 3}}} \right) + \frac{5}{3} \times \left( {\frac{3}{{3 \times 6}} + \frac{3}{{6 \times 9}}} \right) + \frac{5}{9} \times \left( {\frac{9}{{9 \times 18}} + \frac{9}{{18 \times 27}}} \right)\\
 = 5 \times \left( {1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3}} \right) + \frac{5}{3} \times \left( {\frac{1}{3} - \frac{1}{6} + \frac{1}{6} - \frac{1}{9}} \right) + \frac{5}{9} \times \left( {\frac{1}{9} - \frac{1}{{18}} + \frac{1}{{18}} - \frac{1}{{27}}} \right)\\
 = 5 \times \left( {1 - \frac{1}{3}} \right) + \frac{5}{3} \times \left( {\frac{1}{3} - \frac{1}{9}} \right) + \frac{5}{9} \times \left( {\frac{1}{9} - \frac{1}{{27}}} \right)\\
 = 5 \times \frac{2}{3} + \frac{5}{3} \times \frac{2}{9} + \frac{5}{9} \times \frac{2}{{27}}\\
 = 5 \times \frac{2}{3} \times \left( {1 + \frac{1}{9} + \frac{1}{{3 \times 27}}} \right)\\
 = \frac{{10}}{3} \times \frac{{3 \times 27 + 9 + 1}}{{3 \times 27}}\\
 = \frac{{10}}{3} \times \frac{{91}}{{81}}\\
 = \frac{{910}}{{243}}
\end{array}\]