Question

CMR với mọi n nguyên dương thỉ 3ⁿ⁺² - 2ⁿ⁺² + 3ⁿ - 2ⁿ chia hết cho 10

Answer 1

\(3^{n+2}-2^{n+2}+3^n-2^n\)

\(=3^n^{+2}+3^n-2^{n+1}-2^n\)

\(=3^n.\left(3^2+1\right)-2^n.\left(2^2+1\right)=3^n.10-2^n.5=3^n.10-2^{n-1}.10\)

vì 3ⁿ.10 chia hết cho 10

2^n-1 .10  chia hết cho 10

=> \(3^{n+2}-2^{n+2}+3^n-2^n\)chia hết cho 10(đpcm)

Answer 2

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

                                                 =3ⁿ.10-2^n-1.10

                                                 =(3ⁿ-2^n-1).10 chia hết cho 10 với mọi n nguyên dương

Answer 3

Ta có : \(3^{n+2}-2^{n+2}+3^n-2^n\)

       \(=\left(3^{n+2}+3^n\right)-\left(2^{n+2}+2^n\right)\)

        \(=3^n\left(3^2+1\right)-2^n\left(2^2+1\right)\)

        \(=3^n.10-2^n.5=3^n.10-2^{n-1}.10\)

         \(=10\left(3^n-2^n\right)\)

\(\Rightarrow10\left(3^n-2^n\right)⋮10\) ( vì \(10⋮10\) )

Vậy \(3^{n+2}-2^{n+2}+3^n-2^n⋮10\) vs mọi n nguyên dương.

Answer 4

\(3^{n+2}-2^{n+2}+3^n-2^n\)

\(=3^n.3^2-2^n.2^2+3^n-2^n\)

\(=\left(3^n.9+3^n\right)-\left(2^n.2^2+2^n\right)\)

\(=3^n.\left(9+1\right)-2^n.\left(4+1\right)\)

\(=3^n.10-2^n.5\)

\(=5\left(3^n.2-2^n\right)\)

Vì \(\hept{\begin{cases}3^n.2⋮2\\2^n⋮2\end{cases}}\Rightarrow3^n.2-2^n⋮2\)

mà \(5.\left(3^n.2-2^n\right)⋮5\)

\(\Rightarrow5\left(3^n.2-2^n\right)⋮2\)và \(5\)

hay \(5\left(3^n.2-2^n\right)⋮10\)

hay \(3^{n+2}-2^{n+2}+3^n-2^n⋮10\left(đpcm\right)\)