Question

CMR:

1+3+3²+3³+...+3¹¹ chia hết cho 4

Answer 1

Đặt:

\(A=1+3+3^2+3^3+.....+3^{11}\)

\(A=\left(1+3\right)+\left(3^2+3^3\right)+.....+\left(3^{10}+3^{11}\right)\)

\(A=1\left(1+3\right)+3^2\left(1+3\right)+.....+3^{10}\left(1+3\right)\)

\(A=1.4+3^2.4+....+3^{10}.4\)

\(A=4\left(1+3^2+...+3^{10}\right)\)

\(A⋮4\rightarrowđpcm\)

Answer 2

Đặt : B = 1 + 3 + 3² + 3³ + ........+3¹¹

B = (1+3 ) +(3²+3³)+..........+ (3¹⁰+3¹¹)

B=1.(1+3)+3².(1+3 ) +............+ 3¹⁰ . ( 1+3)

B = 1.4 + 3².4 +.................+ 3¹⁰.4

B = 4.(1+3²+..............+3¹⁰⁾

Mà 4 ^\(⋮\) 4

=>4.(1+3²+.........+3¹⁰) ^\(⋮\)4

Vậy B ^\(⋮\)4