11ⁿ⁺²+12²ⁿ⁺¹

=121.11ⁿ+144ⁿ.12

(133-12).11ⁿ+144ⁿ.12

11ⁿ.133-11ⁿ.12+144ⁿ.12

11ⁿ.133+144ⁿ.12-11ⁿ.12

=11.133+12(144ⁿ-11ⁿ)

Ta cso 144ⁿ-11ⁿ : 144-11=133

11.133: 133

Vậy.........

Ta có:

11^{n + 2} + 12^{2n + 1}

= 11ⁿ . 11² + (12²)ⁿ . 12

= 11ⁿ . 121 + 144ⁿ . 12

= 11ⁿ . (121 + 12) + (144ⁿ - 11ⁿ) . 12

= 11ⁿ . 133 + (144ⁿ - 11ⁿ) . 12

Lại có: 144ⁿ \(\equiv\) 11ⁿ (mod 133)

\(\Rightarrow\) 144ⁿ - 11ⁿ \(⋮\) 133

\(\Rightarrow\) ​144ⁿ - 11ⁿ \(⋮\) 133

\(\Rightarrow\) (144ⁿ - 11ⁿ) . 12 \(⋮\) 133

\(\Rightarrow\) 11ⁿ . 133 + (144ⁿ - 11ⁿ) . 12 \(⋮\) 133

Ta có:
$11ⁿ⁺² + 12²ⁿ⁺¹ + 12^2$
$= 11² x 11ⁿ + 12 x (12²)ⁿ + 12^2$
$= 121 x 11ⁿ + 12 x (12²)ⁿ + 144$
$= (133 - 12) x 11ⁿ + 12 x (12²)ⁿ + 144$
$= 133 x 11ⁿ - 12 x 11ⁿ + 12 x 144ⁿ + 144$
$= 133 x 11ⁿ + 12 x (144ⁿ - 11ⁿ) + 144$
Vì:
$133 x 11ⁿ ⋮ 133$
$12 x (144ⁿ - 11ⁿ) = 12 x (144 - 11) x P = 12 x 133 x P ⋮ 133$
$144$ chia cho $133$ dư $11$
Suy ra: $11ⁿ⁺² + 12²ⁿ⁺¹ + 12²$ chia cho $133$ dư $11$

Ta có:
11ⁿ⁺² + 12²ⁿ⁺¹ + 12²
= 11² x 11ⁿ + 12 x (12²)ⁿ + 12²
= 121 x 11ⁿ + 12 x (12²)ⁿ + 144
= (133 - 12) x 11ⁿ + 12 x (12²)ⁿ + 144
= 133 x 11ⁿ - 12 x 11ⁿ + 12 x 144ⁿ + 144
= 133 x 11ⁿ + 12 x (144ⁿ - 11ⁿ) + 144
Vì:
133 x 11ⁿ ⋮ 133
12 x (144ⁿ - 11ⁿ) = 12 x (144 - 11) x P = 12 x 133 x P ⋮ 133
144 chia cho 133 dư 11
Suy ra: 11ⁿ⁺² + 12²ⁿ⁺¹ + 12² chia cho 133 dư 11

11^n+2 + 12^2n+1

= 121*11^n + 144^n*12

= (133-12)11^n + 144^n*12

= 133*11^n + 12*(144-11)

= 133*11^n + 12*133

= 133(11^n + 12) chia hết cho 133.

\(11^{n+2}+12^{2n+1}=11.2.11^n+12.1.12^{2n}\)

\(=121.11^n+12.144^n\)

\(\left(133-12\right).11^n+12.144^n\)

\(133.11^n+\left(144^n-11^n\right).12=133.11^n+133^n.12\)

133.11^n chia hết cho 133

133^n.12 chia hết cho 133

=> 11^n+2  + 12 ^2n+1 chia hết cho 133

la 133 nhe chac chan 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000% luon neu khong minh se chet

b) Với n=1 thì hiển nhiên đúng.

Giả sử mệnh đề đúng với n=k tức:

11^k+1+12^2k-1 chia hết cho 133

Với n=k+1 thì:

11^k+2+12^2k+1=11^k+1.11+12^2k-1.12²=11(11^k+1+12^2k-1)+133.12^2k-1 luôn luôn chia hết cho 133.

Vậy mệnh đề đúng với n=k+1 => dpcm.

tick nha

bạn bấm vào dòng chữ xanh này nhé chứng minh : 11n+2+122n+1 chia hết cho 133

11^n+2+12^2n+1=11^n.11^2+12^2n.12=11^n.121+(12^2)^n.12=11^n.121+144^n.12=11^n.(133-12).144^n.12=(11^n.133-11^n.12)+144^n.12

=11^n.133+(144^n-11^n).12

Vi 11^n.133 chia het cho 133; 144^n-11^n chia het cho 133

suy ra (144^n-11^n).12 chia het cho 133

Chung to 11^n+2+12^2n+1 chia het cho 133.