Question

B=1+7+7^2+7^3+...+7^119

a.chứng tỏ Bchia hết cho 8và 57

b.rút gọn B

 

Answer 1

a)\(B=1+7+7^2+7^3+...+7^{119}\)

\(B=\left(1+7\right)+7^2\left(1+7\right)+7^4\left(1+7\right)+...+7^{118}\left(1+7\right)\)

\(B=8+7^2\cdot8+7^4\cdot8+...+7^{118}\cdot8\)

\(\Rightarrow B⋮8\)(đpcm)

\(B-1=7+7^2+7^3+...+7^{119}⋮7\)

\(\Rightarrow B⋮1+7\cdot8\Rightarrow B⋮57\)

\(B=1+7+7^2+...+7^{119}\)

\(\Rightarrow B=1+7\left(7+7^2+...+7^{118}\right)\)

\(\Rightarrow B=1+7\left(B-7^{119}\right)\)

\(\Rightarrow B=1+7B-7^{120}\)

\(\Rightarrow6B=7^{120}-1\Rightarrow B=\frac{7^{120}-1}{6}\)