3\5+3\5^4+3\5^7+.............+3\5^100

Đáp án:

`A=\frac{3(5^{102}-1)}{5^{100}.124}`

Giải thích các bước giải:

 Đặt `A=\frac{3}{5}+\frac{3}{5^4}+...+\frac{3}{5^{100}}`

`⇒\frac{1}{5^3}A=\frac{3}{5^4}+\frac{3}{5^7}+....+\frac{3}{5^{103}}`

`⇒A-\frac{1}{125}A=\frac{3}{5}+\frac{3}{5^4}+...+\frac{3}{5^{100}}-(\frac{3}{5^4}+\frac{3}{5^7}+....+\frac{3}{5^{103}}`

`⇒\frac{124}{125}A=\frac{3}{5}-\frac{3}{5^{103}}`

`⇒\frac{124}{125}A=\frac{3.5^{102}-3}{5^{103}}`

`⇒\frac{124}{125}A=\frac{3(5^{102}-1)}{5^{103}}`

`⇒A=\frac{[3(5^{102}-1)].5^3}{5^{103}.124}`

`⇒A=\frac{3(5^{102}-1)}{5^{100}.124}`