Ta có: \(A=\frac{1}{5}+\frac{1}{13}+\frac{1}{14}+\frac{1}{15}+\frac{1}{61}+\frac{1}{62}+\frac{1}{63}\)

\(A=\frac{1}{5}+\left(\frac{1}{13}+\frac{1}{14}+\frac{1}{15}\right)+\left(\frac{1}{62}+\frac{1}{62}+\frac{1}{63}\right)\)

\(A=\frac{1}{5}+\frac{1}{15}.3+\frac{1}{63}.3\)

\(A=\frac{1}{5}+\frac{1}{5}+\frac{1}{21}\)

\(A=\frac{47}{105}\)

Mà: \(\frac{47}{105}< \frac{47}{94}=\frac{1}{2}\)

Nên \(A=\frac{1}{5}+\frac{1}{13}+\frac{1}{14}+\frac{1}{15}+\frac{1}{61}+\frac{1}{62}+\frac{1}{63}< \frac{1}{2}\)

vi 1/62>1/80 ;1/62>1/80:...:1/80=0/80

suy ra 1/61+1/62+1/63+...+1/80>1/80+1/80+1/80+...+1/80

moi ve co 20 so hang

Vì \(\frac{1}{13}+\frac{1}{14}+\frac{1}{15}

ta có : \(\dfrac{1}{61}< \dfrac{1}{60}\\ \dfrac{1}{62}< \dfrac{1}{60}\\ ...\\ \dfrac{1}{100}< \dfrac{1}{60}\\ \Rightarrow\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}+...+\dfrac{1}{100}\left(c\text{ó}40.s\text{ố \dfrac{1}{60} }\right)\\ \Rightarrow\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}+...+\dfrac{1}{100}< \dfrac{40}{60}=\dfrac{2}{5}\)

gọi đó là A đi.

Ta có:

1/13+1/14+1/14< 1/12+1/12+1/12=3/12=1/4

1/61+1/62+1/63< 1/60+1/60+1/60=3/60=1/20

=> 1/5+1/13+1/14+1/15+1/61+1/62+1/63<1/5+1/4+1/20=1/2

=>A< 1/2 (ĐPCM)