Question

cho B=1+4+4²+4³+.....+4¹¹
chứng minh B⋮5 và B⋮21
mọi người giúp mình với ạ mình cần gấp lắm

Answer 1

\(B=1+4+4^2+...+4^{11}\)

\(\Rightarrow B=\left(1+4\right)+\left(4^2+4^3\right)+...+\left(4^{10}+4^{11}\right)\)

\(\Rightarrow B=\left(1+4\right)+4^2\left(1+4\right)+...+4^{10}\left(1+4\right)\)

\(\Rightarrow B=\left(1+4\right)\left(4^2+...+4^{10}\right)\)

\(\Rightarrow B=5\left(4^2+...+4^{10}\right)⋮5\)

\(B=1+4+4^2+...+4^{11}\)

\(\Rightarrow B=\left(1+4+4^2\right)+\left(4^3+4^4+4^5\right)...+\left(4^9+4^{10}+4^{11}\right)\)

\(\Rightarrow B=\left(1+4+4^2\right)+4^3\left(1+4+4^2\right)...+4^9\left(1+4+4^2\right)\)

\(\Rightarrow B=\left(1+4+4^2\right)\left(4^3+...+4^9\right)\)

\(\Rightarrow B=21\left(4^3+...+4^9\right)⋮21\)