Question

Cho S = 5 + 5² + 5³ + .............+ 5²⁰⁰⁶

a) tính S

b) chứng minh s chia hết cho 126

 

Answer 1

a) \(S=5+5^2+5^3+...+5^{2006}\)

\(5S=5^2+5^3+5^4+...+5^{2007}\)

\(5S-S=\left(5^2+5^3+5^4+...+5^{2007}\right)-\left(5+5^2+5^3+...+5^{2006}\right)\)

\(4S=5^{2007}-5\)

→ \(S=\frac{5^{2007}-5}{4}\)

b) \(S=5+5^2+5^3+...+5^{2006}\)

\(=\left(5+5^4\right)+\left(5^2+5^5\right)+...+\left(5^{2003}+5^{2006}\right)\)

\(=5\left(1+5^3\right)+5^2\left(1+5^3\right)+...+5^{2003}\left(1+5^3\right)\)

\(=5\cdot126+5^2\cdot126+...+5^{2003}\cdot126\)

\(=\left(5+5^2+...+5^{2003}\right)\cdot126\) chia hết cho \(126\)

Vậy \(S\) chia hết cho \(126\)