Ta có:`a=1/1^2+1/3^2+1/5^2+1/7^2+...+1/99^2`

`a=1+1/3.3+1/5.5+1/7.7+...+1/99^2`

`⇒a>1(1)`

Lại có:`a=1/1^2+1/3^2+1/5^2+1/7^2+...+1/99^2`

`a=1+1/3.3+1/5.5+1/7.7+...+1/99^2`

Ta có:

`1` giữ nguyên

`1/3.3<1/2.4`

`1/5.5<1/4.6`

`1/7.7<1/6.8`

`...`

`1/99.99<1/98.100`

`⇒a<1+1/2.4+1/4.6+1/6.8+...+1/98.100`

`⇒a<1+1/2.(2/2.4+2/4.6+2/6.8+...+2/98.100)`

`⇒a<1+1/2.(1/2-1/4+1/4-1/6+1/6-1/8+...+1/98-1/100)`

`⇒a<1/2.(1/2-1/100)`

`⇒a<1+1/2.(49)/100`

`⇒a<1+49/200`

Mà `1+49/200<2`

`⇒a<2(2)`

Từ `(1)` và `(2)`

`⇒1<a<2`

`⇒a` không phải số tự nhiên`(đpcm)`