Question

CMR : \(3^{n+2}-2^{n+2}+3^n-2^nchiahếtcho10\left(n>0\right)\)

Answer 1

3ⁿ⁺²-2ⁿ⁺²+3ⁿ-2ⁿ=3ⁿ*9-2ⁿ*4+3ⁿ-2ⁿ=3ⁿ*(9+1)-2ⁿ(4+1)=3ⁿ*10-2ⁿ*5=3ⁿ*10-2^n-1*10=(3ⁿ-2^n-1)*10

Vì 10 chia hết cho 10 nên (3ⁿ-2^n-1)*10 chia hết cho 10 hay 3ⁿ⁺²-2ⁿ⁺²+3ⁿ-2ⁿ=3ⁿ*9-2ⁿ*4+3ⁿ-2ⁿ chia hết cho 10(n>0)

Answer 2

3ⁿ⁺²-2ⁿ⁺²+3ⁿ-2ⁿ=3ⁿ*9-2ⁿ*4+3ⁿ-2ⁿ=3ⁿ*(9+1)-2ⁿ(4+1)=3ⁿ*10-2ⁿ*5=3ⁿ*10-2^n-1*10=(3ⁿ-2^n-1)*10

Vì 10 chia hết cho 10 nên (3ⁿ-2^n-1)*10 chia hết cho 10 hay 3ⁿ⁺²-2ⁿ⁺²+3ⁿ-2ⁿ=3ⁿ*9-2ⁿ*4+3ⁿ-2ⁿ chia hết cho 10(n>0)