CMR : 3/4! + 3/5! + 3/6! + … + 3/100! < 1/3! 02/07/2021 Bởi Hailey CMR : 3/4! + 3/5! + 3/6! + … + 3/100! < 1/3!
Ta có: `3/4! + 3/5! + 3/6! + … + 3/100!` `S=3/(1×2×3×4)+3/(1×2×3×4×5)+3/(1×2×3×4×5×6)+….3/(1×2×3×….×99×100)` `S<3/(2×3)+3/(3×4)+3/(4×5)+ …….+3/(99×100)` `S<3(1/(2×3)+1/(3×4)+1/(4×5)+ …….+1/(99×100))` `S<3(1/2−1/3+1/3−1/4+1/4−…..−1/99+1/99−1/100)` `S<3(1/2−1/100)` `⇒S<3×49/100` `⇒S<1/3!` $#JAW#$ Bình luận
Đặt A bằng tổng dãy số trên. =>A = 3/4! + 3/5!+…..+3/100! =>A < 3/4! + 4/5!+…..+99/100! =>A < 1/3! – 1/4! + 1/4! – 1/5! +…..+ 1/99! – 1/100! =>A < 1/3! – 1/100! =>A < 1/3! Vậy : A < 1/3! Bình luận
Ta có: `3/4! + 3/5! + 3/6! + … + 3/100!`
`S=3/(1×2×3×4)+3/(1×2×3×4×5)+3/(1×2×3×4×5×6)+….3/(1×2×3×….×99×100)`
`S<3/(2×3)+3/(3×4)+3/(4×5)+ …….+3/(99×100)`
`S<3(1/(2×3)+1/(3×4)+1/(4×5)+ …….+1/(99×100))`
`S<3(1/2−1/3+1/3−1/4+1/4−…..−1/99+1/99−1/100)`
`S<3(1/2−1/100)`
`⇒S<3×49/100`
`⇒S<1/3!`
$#JAW#$
Đặt A bằng tổng dãy số trên.
=>A = 3/4! + 3/5!+…..+3/100!
=>A < 3/4! + 4/5!+…..+99/100!
=>A < 1/3! – 1/4! + 1/4! – 1/5! +…..+ 1/99! – 1/100!
=>A < 1/3! – 1/100!
=>A < 1/3!
Vậy : A < 1/3!