Question

Tính S=\(\dfrac{11}{2^2}+\dfrac{11}{3^3}+\dfrac{11}{2^4}+...+\dfrac{11}{2^{2011}}\)

Answer 1

\(S=\dfrac{11}{2^2}+\dfrac{11}{2^3}+\dfrac{11}{2^4}+...+\dfrac{11}{2^{2011}}\)

\(\Rightarrow2S=\dfrac{11}{2}+\dfrac{11}{2^2}+\dfrac{11}{2^3}+...+\dfrac{11}{2^{2010}}\)

\(\Rightarrow2S-S=\left(\dfrac{11}{2}+\dfrac{11}{2^2}+\dfrac{11}{2^3}+...+\dfrac{11}{2^{2010}}\right)\)

\(-\left(\dfrac{11}{2^2}+\dfrac{11}{2^3}+\dfrac{11}{2^4}+...+\dfrac{11}{2^{2011}}\right)\)

\(\Rightarrow S=\dfrac{11}{2}-\dfrac{11}{2^{2010}}=\dfrac{11.2^{2009}}{2^{2010}}-\dfrac{11}{2^{2010}}=\dfrac{11.\left(2^{2009}-1\right)}{2^{2010}}\)