Question

Tính tống S=1+1/2+1/2^3+1/2^5+1/2^7+...+1/2^99+1/2^101

Answer 1

\(S=1+\dfrac{1}{2}+\dfrac{1}{2^3}+\dfrac{1}{2^5}+...+\dfrac{1}{2^{101}}\)

\(\Rightarrow S-1=\dfrac{1}{2}+\dfrac{1}{2^3}+\dfrac{1}{2^5}+...+\dfrac{1}{2^{101}}\)

\(\Rightarrow\dfrac{1}{4}\left(S-1\right)=\dfrac{1}{2^3}+\dfrac{1}{2^5}+\dfrac{1}{2^7}+...+\dfrac{1}{2^{103}}\)

\(\Rightarrow\dfrac{1}{4}\left(S-1\right)-\left(S-1\right)=\dfrac{1}{2^3}+\dfrac{1}{2^5}+\dfrac{1}{2^7}+...+\dfrac{1}{2^{103}}-\dfrac{1}{2}-\dfrac{1}{2^3}-...-\dfrac{1}{2^{101}}\)

\(\Rightarrow\dfrac{3}{4}\left(S-1\right)=\dfrac{1}{2^{103}}\)

\(\Rightarrow S-1=\dfrac{1}{2^{103}}:\dfrac{3}{4}\)

\(\Rightarrow S-1=\dfrac{4}{3.2^{103}}\)

\(\Rightarrow S=\dfrac{4}{3.2^{103}}+1\)

Answer 2

⇒S−1=12+123+125+...+12101⇒S−1=12+123+125+...+12101

⇒14(S−1)−(S−1)=123+125+127+...+12103−12−123−...−12101⇒14(S−1)−(S−1)=123+125+127+...+12103−12−123−...−12101

⇒S−1=12103:34⇒S−1=12103:34

⇒S=43.2103+1