Question

[park jimin]

CMR : 7,5²ⁿ+12,6ⁿ chia hết cho 19

~~~##helpme##~~~

Answer 1

[T.Ps]

#)Giải :

(Làm ngắn gọn)

\(VT=7\left(25^n-6^n\right)+19.6^n\)

Dễ thấy \(25^n-6^n⋮\left(25-16\right)=19\)

\(\Rightarrowđpcm\)

Answer 2

[T.Ps]

#)Giải : 

Đặt \(A=7,5^{2n}+12,6^n\)

Với \(n=0\Rightarrow A\left(0\right)=19⋮19\)

Giả sử \(A\left(n\right)⋮19\)với \(n=k\Rightarrow A\left(k\right)=7.5^{2k}+12.6^k⋮19\)

Ta đi chứng minh \(A\left(n\right)⋮19\)với \(n=k+1\)

\(\Rightarrow A\left(k+1\right)=7.5^{2\left(k+1\right)}+12.6^{k+1}\)

\(\Rightarrow A\left(k+1\right)=7.5^{2k}.5^2+12.6^n.6\)

\(\Rightarrow A\left(k+1\right)=7.5^{2k}.6+7.5^{2k}.19+12.6^n.6\)

\(\Rightarrow6A\left(k\right)+7.5^{2k}.19⋮19\)

\(\Rightarrow7.5^{2n}+12.6^n⋮19\)

\(\Rightarrowđpcm\)