Question

chứng minh rằng: S=5+5²+5³+...+5²⁰⁰⁴ chia hết cho 6; 31;156

Answer 1

a)ta có S=5+5²+5³+...+5²⁰⁰⁴ =(5+5²)+(5³+5⁴)+...+(5²⁰⁰³+5²⁰⁰⁴)

S=5.(1+5)+5³.(1+5)+...+5²⁰⁰³.(1+5)

S=5.6+5³.6+..+5²⁰⁰³+6

S=6.(5+5³+...+5²⁰⁰³)

Vì 6 chia hết cho 6

=> S chia hết cho 6

b)S=5.(1+5+5²)+...+5⁹⁸.(1+5+5²)

S= 5.31+...+5⁹⁸.31

S=31.(5+...+5⁹⁸)

vì 31 chia hết cho 31

=> S chia hết cho 31

c)S=5.(1+5+5²+5³)+...+5⁹⁷.(1+5+5²+5³)

S=5.156+...+5⁹⁷.156

S= 156.(5+...+5⁹⁷)

vì 156 chia hết cho 156

=> S chia hết cho 156

Answer 2

\(S=5+5^2+5^3+...+5^{2004}\)

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

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

\(=6\left(5+5^3+...+5^{2003}\right)\)

Vậy S chia hết cho 6.

\(S=5\left(1+5+5^2\right)+...+5^{2002}\left(1+5+5^2\right)\)

\(=\left(1+5+5^2\right)\left(5+...+5^{2002}\right)\)

\(=31\left(5+...+5^{2002}\right)\)

Vậy S chia hết cho 31.

\(S=5\left(1+5+5^2+5^3\right)+...+5^{2001}\left(1+5+5^2+5^3\right)\)

\(=\left(1+5+5^2+5^3\right)\left(5+...+5^{2001}\right)\)

\(=156\left(5+...+5^{2001}\right)\)

Vậy S chia hết cho 156.