Question

tính a =1 + \(\dfrac{1}{2}\)+\(\dfrac{1}{2^2}\)+\(\dfrac{1}{2^3}\)+\(\dfrac{1}{2^4}\)+....+\(\dfrac{1}{2^{20}}\)+\(\dfrac{1}{2^{21}}\)

Answer 1

\(a=1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\dfrac{1}{2^4}+...+\dfrac{1}{20^{20}}+\dfrac{1}{2^{21}}\)

\(\Rightarrow2a=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{20^{19}}+\dfrac{1}{2^{20}}\)

\(\Rightarrow2a-a=\left(2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{20^{19}}+\dfrac{1}{2^{20}}\right)-1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\dfrac{1}{2^4}+...+\dfrac{1}{20^{20}}+\dfrac{1}{2^{21}}\)

\(\Rightarrow a=2-\dfrac{1}{2^{21}}=\dfrac{2^{22}-1}{2^{21}}\)