Question

cho B=1=5+5² +5³+...+5¹⁰¹ chứng minh rằng B là bội chung của 6 và 31

Answer 1

Ta có: \(B=1+5+5^2+\cdots+5^{101}\)

\(=\left(1+5\right)+\left(5^2+5^3\right)+\cdots+\left(5^{100}+5^{101}\right)\)

\(=\left(1+5\right)+5^2\left(1+5\right)+\cdots+5^{100}\left(1+5\right)\)

\(=6\left(1+5^2+\cdots+5^{100}\right)\) ⋮6

Ta có: \(B=1+5+5^2+\cdots+5^{101}\)

\(=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+\cdots+\left(5^{99}+5^{100}+5^{101}\right)\)

\(=\left(1+5+5^2\right)+5^3\left(1+5+5^2\right)+\cdots+5^{99}\left(1+5+5^2\right)\)

\(=31\left(1+5^3+\cdots+5^{99}\right)\) ⋮31