Đáp án:
`A = 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/20^2`
`text{Vì :}` \(\left\{ \begin{array}{l}\dfrac{1}{2^2} < \dfrac{1}{1.2}\\ \dfrac{1}{3^2} < \dfrac{1}{2 . 3} \\ \dfrac{1}{4^2} < \dfrac{1}{3 . 4} \\ .........\\\dfrac{1}{20^2}<\dfrac{1}{19 . 20}\end{array} \right.\)
`text{Nên :}`
`A < 1/(1 . 2) + 1/(2 . 3) + 1/(3.4) + .... + 1/(19 . 20)`
`-> A < 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/19 - 1/20`
`-> A < 1 + (- 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/19) - 1/20`
`-> A < 1 - 1/20`
`text{Ta thấy :}` `1 - 1/20 < 1`
`-> A < 1 (1)`
$\\$
`text{Mặt khác :}`
`text{Vì :}`\(\left\{ \begin{array}{l}\dfrac{1}{2^2} > \dfrac{1}{2.3}\\ \dfrac{1}{3^2} > \dfrac{1}{3.4} \\ \dfrac{1}{4^2} > \dfrac{1}{4.5} \\ .........\\\dfrac{1}{20^2}>\dfrac{1}{20.21}\end{array} \right.\)
`text{Nên :}`
`-> A > 1/(2 . 3) + 1/(3 . 4) + 1/(4 . 5) + ... + 1/(20 . 21)`
`-> A > 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + .... + 1/20 - 1/21`
`-> A > 1/2 + (- 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + .... + 1/20) - 1/21`
`-> A > 1/2 - 1/21`
`-> A > 19/42`
`text{Ta thấy :}` `19/42 = 0,4 > 0`
`-> A > 0 (2)`
$\\$
`text{Từ (1) và (2)}`
`-> 0 < A < 1`
`->` `text{A không phải là 1 số tự nhiên (đpcm)}`
