Câu 5:So sánh 2 biểu thức sau: P=1+7+7^2+....+7^99+7^100 / 1+7+7^2+...+7^98+7^99 và Q =1+9+9^2+...+9^100/1+9+9^2+...+9^99

Ta có:`P=(1+7+7^2+...+7^99+7^100)/(1+7+7^2+...+7^98+7^99)`

`P=(1+7+7^2+...+7^99)/(1+7+7^2+...+7^98+7^99)+7^100/(1+7+7^2+...+7^98+7^99)`

`P=1+1/(1/7^100+1/7^99+1/7^98+...+1/7^2+1/7)`

Và `Q=(1+9+9^2+...+9^100)/(1+9+9^2+...+9^99)`

`Q=(1+9+9^2+...+9^99)/(1+9+9^2+...+9^98+9^99)+9^100/(1+9+9^2+...+9^98+9^99)`

`Q=1+1/(1/9^100+1/9^99+1/9^98+...+1/9)`

Vì `1=1`

`7^100<9^100=>1/7^100>1/9^100`

`7^99<9^99=>1/7^99>1/9^99`

`7^98<9^98=>1/7^98>1/9^98`

`...`

`7<9=>1/7>1/9`

Do đó `1/7^100+1/7^99+1/7^98+...+1/7>1/9^100+1/9^99+1/9^98+...+1/9`

`=>1+1/(1/7^100+1/7^99+1/7^98+...+1/7)<1+1/(1/9^100+1/9^99+1/9^98+...+1/9)`

`=>P<Q`

Vậy `P<Q`