Question

Cho tổng A gồm 2014 số hạng.

\(A=\frac{1}{19}+\frac{2}{19^2}+\frac{3}{19^3}+...+\frac{2014}{19^{2014}}\)

Tính A

Answer 1

\(A=\frac{1}{19}+\frac{2}{19^2}+...+\frac{2014}{19^{2014}}\)

\(\Rightarrow19A=1+\frac{2}{19}+\frac{3}{19^2}+...+\frac{2014}{19^{2013}}\)

\(\Rightarrow19A-A=\left(1+\frac{2}{19}+\frac{3}{19^2}+...+\frac{2014}{19^{2013}}\right)-\left(\frac{1}{19}+\frac{2}{19^2}+...+\frac{2014}{19^{2014}}\right)\)

\(\Rightarrow18A=1+\left(\frac{1}{19}+\frac{1}{19^2}+...+\frac{1}{19^{2013}}\right)-\frac{2014}{19^{2014}}\)

\(\Rightarrow18A=1+\frac{1-\frac{1}{19^{2013}}}{18}-\frac{2014}{19^{2014}}\)

\(\Rightarrow A=\frac{1+\frac{1-\frac{1}{19^{2013}}}{18}-\frac{2014}{19^{2014}}}{18}\)

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