Đáp án: `=\frac{7^{2008}-1}{7^{2007}}x^2y^3`
Giải thích các bước giải:
Đặt `S=1-\frac{1}{7}+\frac{1}{7^2}-\frac{1}{7^3}+.....-\frac{1}{7^{2007}}`
`⇒7S=7-1+\frac{1}{7}-\frac{1}{7^2}+.....-\frac{1}{7^{2006}}`
`⇒7S+S=(1-\frac{1}{7}+\frac{1}{7^2}-\frac{1}{7^3}+.....-\frac{1}{7^{2007}})+(7-1+\frac{1}{7}-\frac{1}{7^2}+.....-\frac{1}{7^{2006}})`
`⇒8S=7-\frac{1}{7^{2007}}=\frac{7^{2008}-1}{7^{2007}}`
`⇒S=\frac{7^{2008}-1}{8.7^{2007}}`
Đặt:
`A=(1-\frac{1}{7}+\frac{1}{7^2}-\frac{1}{7^3}+.....-\frac{1}{7^{2007}})xy^2`
$B=8xy$
`⇒AB=(1-\frac{1}{7}+\frac{1}{7^2}-\frac{1}{7^3}+.....-\frac{1}{7^{2007}})xy^2.8xy`
`=8S.(xy^2.xy)=(8.\frac{7^{2008}-1}{8.7^{2007}}).x^2y^3`
`=\frac{7^{2008}-1}{7^{2007}}x^2y^3`
