Question

Chung to phep tinh ay chia het cho 13

1+3+3²+3³+...+3¹¹⁸+3¹¹⁹

Answer 1

Ta có : S = ( 1 + 3 + 3² ) + ( 3³ + 3⁴ + 3⁵ ) + .... + ( 3¹¹⁴ + 3¹¹⁵ + 3¹¹⁶ ) + ( 3¹¹⁷ + 3¹¹⁸ + 3¹¹⁹ )

              = ( 1 + 3 + 3² ) + ( 3³.1 + 3³.3 + 3³.3² ) + .... + ( 3¹¹⁴.1 + 3¹¹⁴.3 + 3¹¹⁴.3² ) + ( 1.3¹¹⁷ + 3.3¹¹⁷ + 3².3¹¹⁷ )

              = ( 1 + 3 + 3² ) + 3³(1 + 3 + 3²) + .... + 3¹¹⁴(1 + 3 + 3²) + 3¹¹⁷( 1 + 3 + 3² )

              = 13 + 3³.13 + .... + 3¹¹⁴.13 + 3¹¹⁷.13

              = 13( 1 + 3³ + ... + 3¹¹⁴ + 3¹¹⁷ ) chia hết cho 13

Vậy S chia hết cho 13 ( đpcm )

Answer 2

Ta có : S = ( 1 + 3 + 3 2 ) + ( 3 3 + 3 4 + 3 5 ) + .... + ( 3 114 + 3 115 + 3 116 ) + ( 3 117 + 3 118 + 3 119 )

= ( 1 + 3 + 3 2 ) + ( 3 3 .1 + 3 3 .3 + 3 3 .3 2 ) + .... + ( 3 114 .1 + 3 114 .3 + 3 114 .3 2 ) + ( 1.3 117 + 3.3 117 + 3 2 .3 117 )

= ( 1 + 3 + 3 2 ) + 3 3 (1 + 3 + 3 2 ) + .... + 3 114 (1 + 3 + 3 2 ) + 3 117 ( 1 + 3 + 3 2 )

= 13 + 3 3 .13 + .... + 3 114 .13 + 3 117 .13 = 13( 1 + 3 3 + ... + 3 114 + 3 117 ) chia hết cho 13

Vậy S chia hết cho 13 ( đpcm )