Question

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A=1+\(\dfrac{3}{2^3}\)+\(\dfrac{4}{2^4}\)+\(\dfrac{5}{2^5}\)+.....+\(\dfrac{100}{2^{100}}\)

Answer 1

\(A=1+\dfrac{3}{2^3}+\dfrac{4}{2^4}+..+\dfrac{100}{2^{100}}\\ \Rightarrow2A=2+\dfrac{3}{2^2}+\dfrac{4}{2^3}+...+\dfrac{100}{2^{99}}\\ \Rightarrow A=\dfrac{7}{4}+\dfrac{1}{2^3}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}\\ B=\dfrac{1}{2^3}+...+\dfrac{1}{2^{99}}\\ \Rightarrow2B=\dfrac{1}{2^2}+...+\dfrac{1}{2^{98}}\\ \Rightarrow B=\dfrac{1}{4}-\dfrac{1}{2^{99}}\\ \Rightarrow A=\dfrac{7}{4}+\dfrac{1}{4}-\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}\\ =2-\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}\)