Rút gọn: A = 1 + 1/2 + 1/2^2 + 1/2^3 + . . . + 1/2^2012

Đề bài: Rút gọn: $A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\mathrm{...}+\frac{1}{2^{2012}}$ .

Trả lời

Hướng dẫn giải:

Ta có: $A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\mathrm{...}+\frac{1}{2^{2012}}$

Suy ra $2A=2+1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\mathrm{...}+\frac{1}{2^{2011}}$ .

Ta có: $2A-A=\left(2+1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\mathrm{...}+\frac{1}{2^{2011}}\right)-\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\mathrm{...}+\frac{1}{2^{2012}}\right)$

$\iff A=2-\frac{1}{2^{2012}}$.

Vậy $A=2-\frac{1}{2^{2012}}=\frac{2^{2013}}{2^{2012}}-\frac{1}{2^{2012}}=\frac{2^{2013}-1}{2^{2012}}$ .