Câu hỏi của Bùi Thị Phương Anh

Question

Chứng minh tổng sau chia hết cho 3

A = 2 + 2$^2$ + 2$^3$ + ... + 2$^{200}$

Nguyễn Nam · Accepted Answer

$A=2+2^2+2^3+2^4+....+2^{199}+2^{200}$

$\Leftrightarrow A=\left(2+2^2\right)+\left(2^3+2^4\right)+....+\left(2^{199}+2^{200}\right)$

$\Leftrightarrow A=2\left(1+2\right)+2^3\left(1+2\right)+....+2^{199}\left(1+2\right)$

$\Leftrightarrow A=2.3+2^3.3+....+2^{199}.3$

$\Leftrightarrow A=3\left(2+2^3+2^5+....+2^{199}\right)⋮3\left(dpcm\right)$Câu trả lời: $A=2+2^2+2^3+2^4+....+2^{199}+2^{200}$

$\Leftrightarrow A=\left(2+2^2\right)+\left(2^3+2^4\right)+....+\left(2^{199}+2^{200}\right)$

$\Leftrightarrow A=2\left(1+2\right)+2^3\left(1+2\right)+....+2^{199}\left(1+2\right)$

$\Leftrightarrow A=2.3+2^3.3+....+2^{199}.3$

$\Leftrightarrow A=3\left(2+2^3+2^5+....+2^{199}\right)⋮3\left(dpcm\right)$

Ôn tập chương I