Question

chứng tỏ rằng

1+5+5²+.....+5⁴⁰²+5⁴⁰³+5⁴⁰⁴ chia hết cho 31

Answer 1

      \(1+5+5^2+...+5^{404}\)

\(=5^3\left(1+5+5^2\right)+5^4\left(1+5+5^2\right)+...+5^{404}\left(1+5+5^2\right)\)

\(=\left(1+5+5^2\right)\left(5^3+5^4+...+5^{403}+5^{404}\right)\)

\(=31.\left(5^3+5^4+...+5^{403}+5^{404}\right)\)

Vậy tổng trên chia hết cho 31

Answer 2

1 + 5 + 5² + .... + 5⁴⁰⁴

= ( 1 + 5 ) + ( 5² + 5³ ) + ... + ( 5⁴⁰³ + 5⁴⁰⁴ )

= 6 + 5² . ( 1 + 5 ) + ... + 5⁴⁰³ . ( 1 + 5 )

=6 + 5² . 6 + ... + 5⁴⁰³ . 6

= 6 . ( 1 + 5² + ... + 5^{403 )}

= 3 . 2 . ( 1 + 5² + .... + 5⁴⁰³ ) chia hét cho 3 

Answer 3

Ta có: 1 + 5 + 5² + .... + 5⁴⁰⁴

= ( 1 + 5 ) + ( 5² + 5³ ) + ... + ( 5⁴⁰³ + 5⁴⁰⁴ )

= 6 + 5² . ( 1 + 5 ) + ... + 5⁴⁰³ . ( 1 + 5 )

=6 + 5² . 6 + ... + 5⁴⁰³ . 6

= 6 . ( 1 + 5² + ... + 5⁴⁰³ ) chia hết cho 3