Question

 A) Cho M = 1+3+3²+3³+3⁴+.....+3⁹.  Tìm số tự nhiên n biết 2.M +1 = 3ⁿ

B) cho A = 1+4²+4³+4⁴+.....+4⁹⁹

 Tìm x biết: 3A + 13 = 4ⁿ (đây là cùng phần B luôn nhé)

 

Answer 1

A)\(M=1+3+3^2+...+3^9\)\(\Rightarrow3M=3+3^2+3^3+...+3^{10}\)\(\Rightarrow3M-M=\left(3+3^2+3^3+...+3^{10}\right)-\left(1+3+3^2+...+3^9\right)\)

\(\Rightarrow2M=3^{10}-1\)\(\Rightarrow2M+1=3^{10}\)\(\Rightarrow n=10\)

B) \(A=1+4^2+...+4^{99}\)\(\Rightarrow4A=4+4^3+4^4+...+4^{100}\)\(\Rightarrow4A-A=\left(4+4^3+4^4+...+4^{100}\right)-\left(1+4^2+...+4^{99}\right)\)

\(\Rightarrow3A=4^{100}+4-4^2-1\Rightarrow3A=4^{100}-13\Rightarrow3A+13=4^{100}\Rightarrow n=100\)