`S=(-1/7)^0+(-1/7)^1+(-1/7)^2+...+(-1/7)^2007`

Đáp án:

$S=\dfrac{7^{2008}-1}{8.7^{2007}}.$

Giải thích các bước giải:

$S=\left(-\dfrac{1}{7}\right)^0+\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+\dots+\left(-\dfrac{1}{7}\right)^{2007}\\ =1-\dfrac{1}{7}+\left(\dfrac{1}{7}\right)^2-\dots-\left(\dfrac{1}{7}\right)^{2007}\\ 7S=7-1+\dfrac{1}{7}-\dots-\left(\dfrac{1}{7}\right)^{2006}\\ S+7S=\left(1-\dfrac{1}{7}+\left(\dfrac{1}{7}\right)^2-\dots-\left(\dfrac{1}{7}\right)^{2007}\right)+\left(7-1+\dfrac{1}{7}-\dots-\left(\dfrac{1}{7}\right)^{2006}\right)\\ \Leftrightarrow 8S=7-\left(\dfrac{1}{7}\right)^{2007}\\ \Leftrightarrow 8S=7-\dfrac{1}{7^{2007}}\\ \Leftrightarrow 8S=\dfrac{7^{2008}-1}{7^{2007}}\\ \Leftrightarrow S=\dfrac{7^{2008}-1}{8.7^{2007}}.$

`S=(-1/7)^0+(-1/7)^1+(-1/7)^2+...+(-1/7)^2007`

`=>S=1-1/7+(1/7)^2-...-(1/7)^2007`

`=>7S=7-1+1/7-...-(1/7)^2006`

`=>S+7S=(1-1/7+(1/7)^2-...-(1/7)^2007)+(7-1+1/7-...-(1/7)^2006)`

`=>8S=7-(1/7)^2007`

`=>8S=7-\frac{1}{7^2007}`

`=>8S=\frac{7^2008-1}{7^2007}`

`=>S=\frac{7^2008-1}{8.7^2007}`