Question

a) Chứng minh: B = 3¹ + 3² + 3³ + 3⁴ + … + 3²⁰¹⁰ chia hết cho 4.

 b) Chứng minh: C = 5¹ + 5² + 5³ + 5⁴ + … + 5²⁰¹⁰ chia hết cho 31.

 c) Cho S=17+5²+5³+5⁴+ ... +5²⁰¹⁰ . Tìm số dư khi chia S cho 31.

Answer 1

\(B=3+3^2+3^3+3^4+...+3^{2009}+3^{2010}\)

\(=\left(3+3^2\right)+\left(3^3+3^4\right)+...+\left(3^{2009}+3^{2010}\right)\)

\(=3\left(1+3\right)+3^3\left(1+3\right)+...+3^{2009}\left(1+3\right)\)

\(=4.\left(3+3^3+...+3^{2009}\right)\)

⇒ \(B\) ⋮ 4

Answer 2

b: \(C=5\left(1+5+5^2\right)+...+5^{2008}\left(1+5+5^2\right)=31\cdot\left(5+...+5^{2008}\right)⋮31\)