a)Vì
`1/{2×2}< 1/{1×2}`
`1/{3×3}< 1/{2×3}`
`1/{4×4}< 1/{3×4}`
`....`
`1/{2021×2021}< 1/{2020×2021}`
Nên:
`A<1/{1×2} + 1/{2×3}+ 1/{3×4}+ ...+ 1/{2020×2021}`
`A<2-1/{1×2} + 3-2/{2×3}+ 4-3/{3×4}+ ...+ 2021-2020/{2020×2021}`
`A<2/{1×2}- 1/{1×2} + 3/{2×3}- 2/{2×3}+ 4/{3×4}- 3/{3×4}+ ...+ 2021/{2020×2021}- 2020/{20210×2021}`
`A<1-1/2+1/2-1/3+1/3-1/4+...+1/2020-1/2021`
`A<1-1/2021`
`A<2020/2021`
Vì `2020/2021< 1`
Nên `A<2020/2021 < 1`
B)Vì
`1/{2×2}= 1/{1×2}`
`1/{3×3}< 1/{2×3}`
`1/{4×4}< 1/{3×4}`
`....`
`1/{2021×2021}< 1/{2020×2021}`
Nên:
`A<1/{2×2} + 1/{2×3}+ 1/{3×4}+ ...+ 1/{2020×2021}`
`A<1/4 + 3-2/{2×3}+ 4-3/{3×4}+ ...+ 2021-2020/{2020×2021}`
`A<1/4 + 3/{2×3}- 2/{2×3}+ 4/{3×4}- 3/{3×4}+ ...+ 2021/{2020×2021}- 2020/{20210×2021}`
`A<1/4 +1/2-1/3+1/3-1/4+...+1/2020-1/2021`
`A<1/4+ 1/2 -1/2021`
`A<3/4-2020/2021`
Vì `3/4- 1/2021< 3/4`
Nên `A< 3/4`
c)Ta có:
Vì
`1/{2×2}= 1/{2×2}`
`1/{3×3}= 1/{3×3}`
`1/{4×4}< 1/{3×4}`
`....`
`1/{2021×2021}< 1/{2020×2021}`
Nên:
`A<1/{2×2} + 1/{3×3}+ (1/{3×4}+ ...+ 1/{2020×2021})`
`A<1/4+ 1/9+ (4-3/{3×4}+ ...+ 2021-2020/{2020×2021})`
`A<1/4+ 1/9+ (4/{3×4}- 3/{3×4}+ ...+ 2021/{2020×2021}- 2020/{20210×2021})`
`A<1/4+ 1/9+(1/3-1/4+...+1/2020-1/2021)`
`A<1/4+ 1/9+1/3-1/2021`
`A<13/36+1/3-1/2021`
`A< 25/36-1/2021`
Ta có:
`25/36= 25/36`
`13/18= {13×2}/{18×2}= 26/36`
Vì `25/36<26/36`
Nên `25/36< 13/18`
Nên `25/36- 1/2021< 13/18`
Mà `A<25/36- 1/2021`
Nên `A<13/18`
Vậy `A< 13/18`
