Question

[THI MIEU NGUYEN]

a) Cho A = 2 + 2² + 2³ +2⁴

... + 2⁶⁰
Chứng tỏ A \(⋮\)3 ; 7 và 15
b) Cho B = 1 + 5 + 5² + 5³ + ... + 5⁹⁶ + 5⁹⁷ + 5⁹⁸

. Chứng tỏ rằng B \(⋮\)31

Answer 1

[Yuri Sweet[𝕿𝖊𝖆𝖒 𝕹𝖊𝖕𝖆𝖑]]

a)

•  \(A=2+2^2+2^3+...+2^{60}\)

\(=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{59}\left(1+2\right)\)

\(=2.3+2^3.3+...+2^{59}.3\)

\(=3\left(2+2^3+...+2^{59}\right)⋮3\)

• \(A=2+2^2+2^3+...+2^{60}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)

\(=2.7+2^4.7+...+2^{58}.7\)

\(=7\left(2+2^4+2^{58}\right)⋮7\)

• \(A=2+2^2+2^3+...+2^{60}\)​

\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)

\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+...+2^{57}\left(1+2+2^2+2^3\right)\)

\(=2.15+2^5.15+...+2^{57}.15\)

\(=15\left(2+2^5+2^{57}\right)⋮15\)

Answer 2

[Yuri Sweet[𝕿𝖊𝖆𝖒 𝕹𝖊𝖕𝖆𝖑]]

b) \(B=1+5+5^2+5^3+...+5^{96}+5^{97}+5^{98}\)

\(=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+...+\left(5^{96}+5^{97}+5^{98}\right)\)

\(=\left(1+5+5^2\right)+5^3\left(1+5+5^2\right)+..+5^{96}\left(1+5+5^2\right)\)

\(=31+5^3.31+...+5^{96}.31\)

\(=31\left(1+5^3+...+5^{96}\right)⋮31\)