Đáp án:
\(\begin{array}{l}
a)\,\,\,A = \frac{{{3^{2011}} - 3}}{2}\\
b)\,\,x = 2011
\end{array}\)
Giải thích các bước giải:
\(A = {3^1} + {3^2} + {3^3} + .... + {3^{2010}}\)
a) Thu gọn A:
Ta có: \(3A = 3\left( {{3^1} + {3^2} + {3^3} + ...... + {3^{2010}}} \right) = {3^2} + {3^3} + {3^4} + .... + {3^{2010}}\)
\(\begin{array}{l} \Rightarrow 3A - A = {3^2} + {3^3} + {3^4} + .... + {3^{2011}} - \left( {{3^1} + {3^2} + {3^3} + .... + {3^{2010}}} \right)\\ \Rightarrow 2A = {3^{2011}} - {3^1} = {3^{2011}} - 3\\ \Rightarrow A = \frac{{{3^{2011}} - 3}}{2}.\end{array}\)
b) Ta có:
\(\begin{array}{l}2A + 3 = {3^x}\\ \Leftrightarrow 2.\frac{{{3^{2011}} - 3}}{2} + 3 = {3^x}\\ \Leftrightarrow {3^{2011}} - 3 + 3 = {3^x}\\ \Leftrightarrow {3^{2011}} = {3^x}\\ \Leftrightarrow x = 2011.\end{array}\)
