Question

rút gọn biểu thức A=1+1/2+1/2^2+1/2^3+...+1/2^2012

 

Answer 1

`#lv`

`A=1+1/2+1/(2^2)+1/(2^3)+....+1/(2^2012)`

`2A=2+1+1/2+1/(2^2)+...+1/(2^2011)`

`2A-A=(2+1+1/2+1/(2^2)+...+1/(2^11))-(1+1/2+1/(2^2)+1/(2^3)+...+1/(2^2012))`

`A=2-1/(2^2012)`

Vậy .... 

Answer 2

\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2012}}\\ 2A=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2011}}\\ 2A-A=\left(2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2011}}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2012}}\right)\\ A=2-\dfrac{1}{2^{2012}}\\ A=\dfrac{2^{4024}-1}{2^{2012}}\)