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\(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{99}}\)
\(\Leftrightarrow3A=1+\frac{1}{3}+\frac{1}{3^{^2}}+...+\frac{1}{3^{98}}\)
\(\Leftrightarrow3A-A=1-\frac{1}{3^{99}}\)
\(\Leftrightarrow2A=1-\frac{1}{3^{99}}\)
\(\Leftrightarrow A=\left(1-\frac{1}{3^{99}}\right)\div2\)
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)
\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(A=1-\frac{1}{100}\)
\(A=\frac{99}{100}< 2\left(đpcm\right)\)
a)2A=1+1/2^1+1/2^2+...+1/2^98
2A-A=1+1/2^1+1/2^2+...+1/2^98-(1/2^1+1/2^2+...+1/2^99)
A=1+1/2^1+1/2^2+...+1/2^98-1/2^1-1/2^2-...-1/2^99
A=1-1/2^99