Question

1+1/2+1/4+1/8+...+1/1024

Answer 1

\(A=1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+....+\dfrac{1}{1024}\)

\(2A=2+1+\dfrac{1}{2}+\dfrac{1}{4}+....+\dfrac{1}{512}\)

\(2A-A=\left(2+1+\dfrac{1}{2}+\dfrac{1}{4}+....+\dfrac{1}{512}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+....+\dfrac{1}{1024}\right)\)

\(A=2-\dfrac{1}{1024}\)

\(A=\dfrac{2047}{1024}\)

Answer 2

\(A=1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{1024}\\ =1+\dfrac{1}{2^1}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{10}}\\ \Rightarrow\dfrac{1}{2}A=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{11}}\\ \Rightarrow A-\dfrac{1}{2}A=1-\dfrac{1}{2^{11}}\\ \Rightarrow A=2-\dfrac{1}{2^{10}}\)

Answer 3

   A =        1 + \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{8}\)+.....+\(\dfrac{1}{512}\)+ \(\dfrac{1}{1024}\)

2A =  2 + 1 + \(\dfrac{1}{2}\)  + \(\dfrac{1}{4}\)+ \(\dfrac{1}{8}\)+....+ \(\dfrac{1}{512}\)

2A - A = 2 - \(\dfrac{1}{1024}\)

A         = 2 - \(\dfrac{1}{1024}\)

A = \(\dfrac{2047}{1024}\)