Question

Cho \(A=1+5+5^1+5^2+5^3+...+5^{71}\)

a, Tìm \(x\) biết \(4.A+x=5^{72}\)

b, Chứng minh rằng \(A\) chia hết cho \(31\)

cUốI tUầN eM nỘp

Answer 1

b) \(A=1+5+5^1+5^2+5^3+...+5^{71}\)

\(\Rightarrow A=\left(1+5^1+5^2\right)+5^3\left(1+5^1+5^2\right)+...+5^{69}\left(1+5^1+5^2\right)\)

\(\Rightarrow A=31+5^3.31+...+5^{69}.31\)

\(\Rightarrow A=31\left(1+5^3+...+5^{69}\right)⋮31\left(dpcm\right)\)

Answer 2

a) \(A=1+5^1+5^2+5^3+...+5^{71}\)

\(\Rightarrow A=\dfrac{5^{71+1}-1}{5-1}=\dfrac{5^{72}-1}{4}\)

\(4A+x=5^{72}\)

\(\Rightarrow4.\dfrac{5^{72}-1}{4}+x=5^{72}\)

\(\Rightarrow5^{72}-1+x=5^{72}\)

\(\Rightarrow x=1\)