$S=2^3+2^{23}+2^{43}+...+2^{40023}$
$S=2^3(1+2^{20}+2^{40}+...+2^{40020})$
$S=2^3\begin{bmatrix} (2^{20})^0+(2^{20})^1+(2^{20})^2+...+(2^{20})^{2001} \end{bmatrix}$
Đặt $A=(2^{20})^0+(2^{20})^1+(2^{20})^2+...+(2^{20})^{2001}$
$\Rightarrow 2^{20}A=(2^{20})^1+(2^{20})^2+(2^{20})^3+...+(2^{20})^{2002}$
$2^{20}A-A=(2^{20})^1+(2^{20})^2+(2^{20})^3+...+(2^{20})^{2002}-(2^{20})^0+(2^{20})^1+(2^{20})^2+...+(2^{20})^{2001}$
$(2^{20}-1)A=(2^{20})^{2002}-(2^{20})^0$
$A=\frac{(2^{20})^{2002}-(2^{20})^0}{2^{20}-1}$
$A=\frac{2^{20024}-1}{2^{20}-1}$
$\Rightarrow S=\frac{2^3(2^{40024}-1)}{2^{20}-1}$
$S=\frac{2^{40027}-2^3}{2^{20}-2^3}$
Bài viết đã được chỉnh sửa nội dung bởi kingkn02: 03-06-2014 - 07:52