Question

[Trần Thanh Quyên]

chứng minh rẳng 1/5 + 1/5^2 + 1/5^3 +..+1/5^n < 1/4

Answer 1

[soyeon_Tiểu bàng giải]

Đặt \(A=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+\frac{1}{5^n}\)

\(5A=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{n-1}}\)

\(5A-A=\left(1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{n-1}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+\frac{1}{5^n}\right)\)

\(4A=1-\frac{1}{5^n}< 1\)

=> \(A< \frac{1}{4}\left(đpcm\right)\)

Answer 2

[o0o I am a studious person o0o]

Gọi dãy số trên là : A 

Ta có : \(A=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+......+\frac{1}{5^n}\)

\(\Rightarrow5A=1+\frac{1}{5}+\frac{1}{5^2}+......+\frac{1}{5^{n-1}}\)

\(\Rightarrow5A-A=\left(1+\frac{1}{5^2}+.....+\frac{1}{5^{n-1}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+.....+\frac{1}{5^n}\right)\)

\(\Rightarrow4A=1-\frac{1}{5^n}< 1\)

\(\Rightarrow4A< 1\Rightarrow A< \frac{1}{4}\)