Đáp án:
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Giải thích các bước giải:
`1`
`1/3+1/15+1/35+...+1/49.51`
`=1/1.3+1/3.5+1/5.7+...+1/49.51`
`=1-1/3+1/3-1/5+1/5-1/7+...+1/49-1/51`
`=1-1/51`
`=50/51`
`2`
`1/6+1/12+1/20+...+1/(n(n+1))=49/100`
`1/2.3+1/3.4+1/4.5+...+1/(n(n+1))=49/100`
`1/2-1/3+1/3-1/4+1/4-1/5+...+1/n-1/(n+1)=49/100`
`1/2-1/(n+1)=49/100`
`1/(n+1)=1/2-49/100`
`1/(n+1)=1/100`
`n+1=100`
`n=100-1`
`n=99`
Vậy `n=99`
`3`
`2/15+2/35+2/63+...+2/((2n-1).(2n+1))`
`2/3.5+2/5.7+2/7.9+...+2/((2n-1).(2n+1))`
`1/3-1/5+1/5-1/7+1/7-1/9+...+1/(2n-1)-1/(2n+1)`
`1/3-1/(2n+1)`
Mà `1/3-1/(2n+1)<1/3` nên:
`2/15+2/35+2/63+...+2/((2n-1).(2n+1))<1/3`
`4`
Sai đề
`5`
`(1-1/4).(1-1/9).(1-1/16)*...*(1-1/10000)`
`=3/4*8/9*15/16*...*9999/10000`
`=1.3/2^2*2.4/3^2*3.5/4^2*...*99.101/100^2`
`=(1.2.3.....99)/(2.3.4.....100)*(3.4.5.....101)/(2.3.4.....100)`
`=1/100*101/2`
`=101/200`
