Cho A = 1/1.2 + 1/3.4 + ... + 1/37.38 và B = 1/20.38 + 1/21.27 + ... + 1/38.20. Chứng minh A/B là số nguyên

Giải thích các bước giải:

Ta có:

$A=\dfrac{1}{1.2}+\dfrac{1}{3.4}+...+\dfrac{1}{37.38}$

$\to A=\dfrac{2-1}{1.2}+\dfrac{4-3}{3.4}+...+\dfrac{38-37}{37.38}$

$\to A=\dfrac11-\dfrac12+\dfrac13-\dfrac14+...+\dfrac1{37}-\dfrac1{38}$

$\to A=(\dfrac11+\dfrac13+...+\dfrac1{37})-(\dfrac12+\dfrac14+...+\dfrac1{38})$

$\to A=(\dfrac11+\dfrac13+...+\dfrac1{37})+(\dfrac12+\dfrac14+...+\dfrac1{38})-2(\dfrac12+\dfrac14+...+\dfrac1{38})$

 $\to A=\dfrac11+\dfrac12+...+\dfrac1{37}+\dfrac{1}{38}-(1+\dfrac12+...+\dfrac1{19})$

$\to A=\dfrac1{20}+\dfrac1{21}+...+\dfrac1{38}$

Ta có:

$B=\dfrac1{20.38}+\dfrac1{21.37}+...+\dfrac{1}{38.20}$

$\to B=\dfrac1{20.38}+\dfrac1{21.37}+...+\dfrac{1}{29.29}+...+\dfrac{1}{38.20}$

$\to B=\dfrac2{20.38}+\dfrac2{21.37}+...+\dfrac{2}{28.30}+\dfrac{1}{29.29}$

$\to B=2(\dfrac1{20.38}+\dfrac1{21.37}+...+\dfrac{1}{28.30})+\dfrac{1}{29.29}$

$\to 58B=2(\dfrac{58}{20.38}+\dfrac{58}{21.37}+...+\dfrac{58}{28.30})+\dfrac{58}{29.29}$

$\to 58B=2(\dfrac{20+38}{20.38}+\dfrac{21+37}{21.37}+...+\dfrac{28+30}{28.30})+\dfrac{29+29}{29.29}$

$\to 58B=2(\dfrac{1}{20}+\dfrac{1}{38}+\dfrac1{21}+\dfrac{1}{37}+..+\dfrac{1}{28}+\dfrac1{30})+\dfrac{2}{29}$

$\to 58B=2(\dfrac{1}{20}+\dfrac{1}{38}+\dfrac1{21}+\dfrac{1}{37}+..+\dfrac{1}{28}+\dfrac1{30}+\dfrac{1}{29})$

$\to 58B=2(\dfrac{1}{20}+\dfrac1{21}+...+\dfrac{1}{38})$

$\to 29B=\dfrac{1}{20}+\dfrac1{21}+...+\dfrac{1}{38}$

$\to 29B=A$

$\to \dfrac{A}{B}=29\in Z$