`1.`
` A = 1/3.5 + 1/5.7 + 1/7.9 +...+1/37.39`
`A=1/3 - 1/5 + 1/5 - 1/7 +1/7 - 1/9 +... + 1/37 - 1/39`
`A=1/3 - 1/39 `
`A=4/13 `
`2.`
`B=1/3.4 + 1/4.5 + 1/5.6 +...+1/95.96 `
`B=1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 +....+1/95 - 1/96 `
`B=1/3 - 1/96`
`B=31/96 `
`3.`
`(1-1/2)(1-1/3)(1-1/4)...(1-1/2011)`
`= 1/2 . 2/3 . 3/4....2010/2011`
`= (1.2.3....2010)/(2.3.4...2011)`
`=1/2011 `
`4.`
` 1 1/2 . 1 1/3 . 1 1/4 . 1 1/5 ... 1 1/999`
`=3/2.(4)/(3) . 5/4 . 6/5 ...1000/999`
`=(3.4.5.6..1000)/(2.3.4.5...999)`
`=1000/2`
`=500`
`5.`
a) để `A` là phân số thì ` n + 2 ` $\neq$ `0`
` => n +2 ` $\neq$ `0`
`=> n ` $ \neq$ `-2`
b) để `A` là số nguyên thì
` 19 \vdots n + 2 `
` n +2 \inƯ(19)={1;-1;19;-19}`
`=>n\in{-1;-3;17;-21}`
`7.`
gọi `ƯCLN(14n+3;21n+5)` là `d`
ta có :\(\left[ \begin{array}{l}14n+3 \vdots d\\21n+5 \vdots d\end{array} \right.\)
\(\left[ \begin{array}{l}3(14n+3 )\vdots d\\2(21n+5) \vdots d\end{array} \right.\)
\(\left[ \begin{array}{l}42n+9\vdots d\\42n+10 \vdots d\end{array} \right.\)
`( 42n+9 - 42n+10) \vdots d `
` -1 \vdots d `
`d \inƯ(-1)={1;-1}`
`d=-1;1`
vậy ps `(14n+3)/(21n+5)` tối giản với mọi `n\inZ`
