Lưu ý : / = phần , ^ = mũ Tính : S = 1/3 + 1/3^2 + 1/3^4 +..+ 1/3^98 + 1/3^100 Cần gấp ạ cảm ơn

Đáp án:

$S=\dfrac{8.3^{99}+3^{100}-1}{8.3^{100}}.$

Giải thích các bước giải:

$S=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^4}+\dots+\dfrac{1}{3^{98}}+\dfrac{1}{3^{100}}\\ A=\dfrac{1}{3^2}+\dfrac{1}{3^4}+\dots+\dfrac{1}{3^{98}}+\dfrac{1}{3^{100}}\\ 3^2.A=1+\dfrac{1}{3^2}+\dots+\dfrac{1}{3^{96}}+\dfrac{1}{3^{98}}\\ 3^2.A-A=\left(1+\dfrac{1}{3^2}+\dots+\dfrac{1}{3^{96}}+\dfrac{1}{3^{98}}\right)-\left(\dfrac{1}{3^2}+\dfrac{1}{3^4}+\dots+\dfrac{1}{3^{98}}+\dfrac{1}{3^{100}}\right)\\ \Leftrightarrow 8A=1-\dfrac{1}{3^{100}}\\ \Leftrightarrow A=\dfrac{1-\dfrac{1}{3^{100}}}{8}\\ S=\dfrac{1}{3}+A\\ =\dfrac{1}{3}+\dfrac{1-\dfrac{1}{3^{100}}}{8}\\ =\dfrac{1}{3}+\dfrac{3^{100}-1}{8.3^{100}}\\ =\dfrac{8.3^{99}}{3.8.3^{99}}+\dfrac{3^{100}-1}{8.3^{100}}\\ =\dfrac{8.3^{99}+3^{100}-1}{8.3^{100}}.$

`S = 1/3 + 1/3^2 + 1/3^4 + ... + 1/3^{98} + 1/3^{100}`

Đặt `: A = 1/3^2 + 1/3^4 + ... + 1/3^{98} + 1/3^{100}`

`=> 3^2 . A = 1 + 1/3^2 + ... + 1/3^{96} + 1/3^{98}`

`=> 9A - A = ( 1 + 1/3^2 + ... + 1/3^{96} + 1/3^{98} ) - ( 1/3^2 + 1/3^4 + ... + 1/3^{98} + 1/3^{100} )`

`=> 8A = 1 - 1/3^{100}`

`=> A = 1/8 - 1/ ( 8 . 3^{100} )`

Ta có `: S  = 1/3 + A`

`=> S = 1/3 + 1/8 - 1 / ( 8 . 3^{100 } )`

`=> S = 11/24 - 1/ ( 8 .3^{100} )`

Vậy `: S = 11/24 - 1/ ( 8 .3^{100} )`