\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+.....+\frac{1}{105}\)

=\(\frac{2}{6}+\frac{2}{12}+\frac{1}{20}+.....+\frac{2}{210}\)

= \(2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+.....+\frac{1}{14.15}\right)\)

= \(2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{14}-\frac{1}{15}\right)\)

= \(2\left(\frac{1}{2}-\frac{1}{15}\right)\)

= 2 . \(\frac{13}{30}\)

= \(\frac{13}{15}\)

nhớ giải chi tiết

\(A=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+....+\frac{1}{105}+\frac{1}{210}\)

 =>   \(\frac{1}{2}A=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+.....+\frac{1}{210}+\frac{1}{240}\)

                 \(=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+.....+\frac{1}{14.15}+\frac{1}{15.16}\)

                 \(=\frac{1}{2}-\frac{1}{3}+\frac{!}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{14}-\frac{1}{15}+\frac{1}{15}-\frac{1}{16}\)

               \(=\frac{1}{2}-\frac{1}{16}=\frac{7}{16}\)

=>   \(A=\frac{7}{8}\)

nhân 2 vế với 1/2 ta có

1/2 x A = 1/2 x (1/3 + 1/6 +1/10 + 1/15 + .......+1/91 + 1/105 )

1/2 x A = 1/6 +1/12 + 1/20 + 1/30 + ...............+1/182 + 1/210

1/2 x A = 1/(2x3) + 1/(3x4) + 1/(4x5) + 1/(5x6) +................+1/(13x14) + 1/(14x15)

1/2 x A = 1/2 - 1/3 +1/3 -1/4 + 1/4 - 1/5  +1/5 - 1/6+.........+1/13 - 1/14 + 1/14 - 1/15

1/2 x A = 1/2 - 1/15 =13/30

=> A = 13/30 : 1/2=13/15 <1 

tách ra

\(1\)) \(5-\left(10-x\right)=7\)
\(10-x=5-7\)
\(10-x=-2\)
\(x=10-\left(-2\right)\)
\(x=12\)

\(2\)) \(-32-\left(x-5\right)=0\)
\(x-5=-32-0\)
\(x-5=-32\)
\(x=-32+5\)
\(x=-27\)

P = \(\dfrac{1}{3}\) + \(\dfrac{1}{6}\) + \(\dfrac{1}{10}\)+.....+ \(\dfrac{1}{105}\)

P = \(\dfrac{2}{2}\) \(\times\) ( \(\dfrac{1}{3}\) + \(\dfrac{1}{6}\) + \(\dfrac{1}{10}\)+ ......+ \(\dfrac{1}{105}\))

P =  2 \(\times\) ( \(\dfrac{1}{6}\) + \(\dfrac{1}{12}\) + \(\dfrac{1}{20}\)+......+ \(\dfrac{1}{210}\))

P = 2 \(\times\) ( \(\dfrac{1}{2\times3}\) + \(\dfrac{1}{3\times4}\) + \(\dfrac{1}{4\times5}\)+.....+\(\dfrac{1}{14\times15}\))

P = 2 \(\times\) ( \(\dfrac{1}{2}-\dfrac{1}{3}\) + \(\dfrac{1}{3}-\dfrac{1}{4}\) + \(\dfrac{1}{4}\) - \(\dfrac{1}{5}\) +......+ \(\dfrac{1}{14}\) - \(\dfrac{1}{15}\))

P = 2  \(\times\) ( \(\dfrac{1}{2}\) - \(\dfrac{1}{15}\))

P = 2 \(\times\) \(\dfrac{13}{30}\)

P = \(\dfrac{13}{15}\)