Đáp án:
Giải thích các bước giải:
Đặt `S=\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+.....+\frac{1}{80}`
`⇒S=(\frac{1}{21}+\frac{1}{22}+.....+\frac{1}{30})+(\frac{1}{31}+.....+\frac{1}{40})+(\frac{1}{41}+.....+\frac{1}{50})+(\frac{1}{61}+.....+\frac{1}{70})+(\frac{1}{71}+.....+\frac{1}{80})`
`>(\frac{1}{30}+\frac{1}{30}+.....+\frac{1}{30})+(\frac{1}{40}+.....+\frac{1}{40})+(\frac{1}{50}+.....+\frac{1}{50})+(\frac{1}{60}+.....+\frac{1}{60})+(\frac{1}{70}+.....+\frac{1}{70})+(\frac{1}{80}+.....+\frac{1}{80})` ($10$ số hạng `\frac{1}{30};\frac{1}{40};.....;\frac{1}{80}`)
`=\frac{10}{30}+\frac{10}{40}+\frac{10}{50}+\frac{10}{60}+\frac{10}{70}+\frac{10}{80}`
`=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}`
`>(\frac{1}{3}+\frac{1}{6})+(\frac{1}{4}+\frac{1}{8})+\frac{1}{5}`
`=\frac{1}{2}+\frac{3}{8}+\frac{1}{5}=0,5+0,375+0,2=1,075>1(1)`
Lại có: `S=\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+.....+\frac{1}{80}`
`=(\frac{1}{21}+\frac{1}{22}+.....+\frac{1}{30})+(\frac{1}{31}+.....+\frac{1}{40})+(\frac{1}{41}+.....+\frac{1}{50})+(\frac{1}{61}+.....+\frac{1}{70})+(\frac{1}{71}+.....+\frac{1}{80})`
`<(\frac{1}{20}+\frac{1}{20}+.....+\frac{1}{20})+(\frac{1}{30}+.....+\frac{1}{30})+(\frac{1}{40}+.....+\frac{1}{40})+(\frac{1}{50}+.....+\frac{1}{50})+(\frac{1}{60}+.....+\frac{1}{60})+(\frac{1}{70}+.....+\frac{1}{70})` ($10$ số hạng `\frac{1}{20};\frac{1}{30};.....;\frac{1}{70}`)
`=\frac{10}{20}+\frac{10}{30}+\frac{10}{40}+\frac{10}{50}+\frac{10}{60}+\frac{10}{70}`
`=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}`
`=(\frac{1}{2}+\frac{1}{3}+\frac{1}{6})+(\frac{1}{4}+\frac{1}{5}+\frac{1}{7})`
`<1+(\frac{1}{3}+\frac{1}{3}+\frac{1}{3})`
`=1+3.\frac{1}{3}=1+1=2(2)`
Từ $(1);(2)⇒1<S<2$
Mà không có số tự nhiên nào nằm giữa $1$ và $2$
$⇒S$ không là số tự nhiên (đpcm)
