Câu hỏi của HKT_Bí Mật

Question

Tính Nhanh

S = 1 + 2 + 2² + 2³ +.....+ 2^63

Phước Lộc · Accepted Answer

$S=1+2+2^2+2^3+...+2^{63}$

$\Rightarrow2S=2+2^2+2^3+2^4+...+2^{64}$

$\Rightarrow2S-S=\left(2+2^2+2^3+2^4+...+2^{64}\right)-\left(1+2+2^2+2^3=...+2^{63}\right)$

$\Rightarrow S=2+2^2+2^3+2^4+...+2^{64}-1-2-2^2-2^3-...-2^{63}$

$\Rightarrow S=\left(2-2\right)+\left(2^2-2^2\right)+\left(2^3-2^3\right)+...+\left(2^{63}-2^{63}\right)+2^{64}-1$

$\Rightarrow S=2^{64}-1$

Câu trả lời:

$S=1+2+2^2+2^3+...+2^{63}$

$\Rightarrow2S=2+2^2+2^3+2^4+...+2^{64}$

$\Rightarrow2S-S=\left(2+2^2+2^3+2^4+...+2^{64}\right)-\left(1+2+2^2+2^3=...+2^{63}\right)$

$\Rightarrow S=2+2^2+2^3+2^4+...+2^{64}-1-2-2^2-2^3-...-2^{63}$

$\Rightarrow S=\left(2-2\right)+\left(2^2-2^2\right)+\left(2^3-2^3\right)+...+\left(2^{63}-2^{63}\right)+2^{64}-1$

$\Rightarrow S=2^{64}-1$