Chứng minh với mọi n€ N* 6^2n+3^n+2+3^n chia hết cho 11
1 câu trả lời
$6^{2n}+3^{n+2}+3^{n}
=6^{2n}-3^{n}+3^{n+2}+3^{n}+3^{n}
=(36^{n}-3^{n})+(3^{n+2}+3^{n}+3^{n})
=3^{n}(12^{n}-1)+3^{n}(3^2+1+1)
=3^{n}(12-1)(12^{n-1}+12^{n-2}...+1)+3^{n}.11
=11.[3^{n}(12^{n-1}+12^{n-2}...+1)+3^{n}] \vdots 11$
Lưu ý $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+...+b^{n-1})$
