Bài 3:
`x/6 = 1/y = 1/2 `
`=> x = 1/2 . 6 và y = 1 : 1/2`
`=> x = 3 và y = 2`
Vậy `x = 3; y = 2`
Bài 4:
`A = 1/(2.3) + 1/(3.4) + ... +1/(19.20) `
`= 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/19 - 1/20`
`= 1/2 - 1/20`
`= 10/20 - 1/20`
`= 9/20`
\(B=\dfrac{1}{7}+\dfrac{1}{91}+\dfrac{1}{247}+\dfrac{1}{475}+\dfrac{1}{775}+\dfrac{1}{1147}\)
\(B=\dfrac{1}{1.7}+\dfrac{1}{7.13}+\dfrac{1}{13.19}+\dfrac{1}{19.25}+\dfrac{1}{25.31}+\dfrac{1}{31.37}\)
\(6B=1-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{25}+\dfrac{1}{25}-\dfrac{1}{31}+\dfrac{1}{31}-\dfrac{1}{37}\)
\(6B=1-\dfrac{1}{37}\)
\(6B=\dfrac{37}{37}-\dfrac{1}{37}\)
\(6B=\dfrac{36}{37}\)
\(B=\dfrac{36}{37}:6\)
\(B=\dfrac{36}{37}.\dfrac{1}{6}\)
\(B=\dfrac{6.1}{37.1}\)
\(B=\dfrac{6}{37}\)
`B = 1/7 + 1/91 + 1/247 + 1/475 + 1/775 + 1/1147`
`= 1/(1.7) + 1/(7.13) + 1/(13.19) + 1/(19.25) + 1/(25.31) + 1/(31.37) `
`= 1/6 . (6/(1.7) + 6/(7.13) + 6/(13.19) + 6/(19.25) + 6/(25.31) + 6/(31.37) )`
`= 1/6 . (1 - 1/7 + 1/7 - 1/13 + 1/13 - 1/19 + 1/19 - 1/25 + 1/25 - 1/31 + 1/31 - 1/37) `
`= 1/6 . (1 - 1/37) `
`= 1/6 . 36/37`
`= 6/37`
Bài 5:
`M = 1/11 + 1/12 + 1/13 + ...+1/19` (Có: `(19 - 11) : 1 + 1 = 9` số hạng)
`< 1/11 + 1/11 + 1/11 + ... + 1/11`
`= 9/11 < 1`
`M = 1/11 + 1/12 + 1/13 + ...+1/19`
`> 1/19 + 1/19 + 1/19 + ... + 1/19`
`= 9/19 > 0`
Nên `0 < M < 1`
Vậy M không là số nguyên
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`N = 1/3 + 1/4 + 1/5 + 2/7 + 2/9 + 2/11`
Ta có:
`1/6 + 1/6 + 1/6 < 1/3 + 1/4 + 1/5 < 1/3 + 1/3 + 1/3`
`=> 3/6 < 1/3 + 1/4 + 1/5 < 3/3 `
`=> 1/2 < 1/3 + 1/4 + 1/5 < 1`
Lại có:
`2/12 + 2/12 + 2/12 < 2/7 + 2/9 + 2/11 < 2/6 + 2/6 + 2/6`
`=> 6/12 < 2/7 + 2/9 + 2/11 < 6/6`
`=> 1/2 < 2/7 + 2/9 + 2/11 < 1`
Khi đó: `1/2 + 1/2 < 1/3 + 1/4 + 1/5 + 2/7 + 2/9 + 2/11 < 1 + 1`
`=> 1 < N < 2`
Vậy N không là số nguyên
