Question

S=4+4^2+4^3+4^4+...+4^60

CMR S chia hết cho 21

 

 

Answer 1

S = 4 + 4² + 4³ + 4⁴ + ...... + 4⁶⁰ 

S = ( 4 + 4² + 4³ ) + ........... + ( 4⁵⁸ + 4⁵⁹ + 4⁶⁰ )

S = 4( 1 + 4 + 4² ) + .......... + 4⁵⁸( 1 + 4 + 4² )

S = 4.21 + ..... + 4⁵⁸.21

S = 21 ( 4 + ..... + 4⁵⁸ )

vì 21 chia hết cho 21 => 21 ( 4 + ....... + 4⁵⁸ ) chia hết cho 21

=> S chia hết cho 21

Answer 2

S = 4 + 4² + 4³ + ... + 4⁶⁰

= ( 4 + 4² + 4³ ) + ( 4⁴ + 4⁵ + 4⁶ ) + ... + ( 4⁵⁸ + 4⁵⁹ + 4⁶⁰ )

= 4 . ( 1 + 4 + 4² ) + 4⁴ . ( 1 + 4 + 4² ) + ... + 4⁵⁸ . ( 1 + 4 + 4² )

= 4 . 21 + 4⁴ . 21 + ... + 4⁵⁸ . 21

= 21 . ( 4 + 4⁴ + ... + 4⁵⁸ ) \(⋮\)21

Suy ra S chia hết cho 21