Question

A=1+5+5²+5³+...+5⁴⁰³+5⁴⁰⁴

CMR A chia hết cho 31

Answer 1

=> A = ( 1 + 5 + 5² ) + ( 5³ + 5⁴ + 5⁵ ) + ... + ( 5⁴⁰² + 5⁴⁰³ + 5⁴⁰⁴ )

=> A = 1.( 1 + 5 + 5.5 ) + 5³.( 1 + 5 + 5.5 ) + .... + 5⁴⁰².( 1 + 5 + 5.5 )

=> A = 1.31 + 5³.31 + .... + 5⁴⁰².31

=> A = 31.( 1 + 5³ + .... + 5⁴⁰² )

Vì 31 ⋮ 31 nên A ⋮ 31 ( đpcm )
 

Answer 2

A=1+5+52+53+...+5403+5404CMR A chia hết cho 31

Answer 3

A=1+5+5²+5³+...+5⁴⁰³+5⁴⁰⁴

\(\Rightarrow\)5*A = 5( 1+5+5²+5³+...+5⁴⁰³+5⁴⁰⁴ )

= 5+5²+5³+...+5⁴⁰³+5⁴⁰⁴+5⁴⁰⁵

\(\Rightarrow\)4*A = ( 5+5²+5³+...+5⁴⁰³+5⁴⁰⁴+5⁴⁰⁵ ) - ( 1+5+5²+5³+...+5⁴⁰³+5⁴⁰⁴ )

= 5⁴⁰⁵ -1

\(\Rightarrow\)A = ( 5⁴⁰⁵ -1 ) :4