Question

cho n là số nguyên dương hãy chứng minh với mọi n thì 3ⁿ⁺² -2ⁿ⁺² +3ⁿ -2ⁿ chia hết cho 10

Answer 1

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

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

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

\(=3^n\cdot10-2^n\cdot5\)

\(=3^n\cdot10-2^{n-1}\cdot2\cdot5\)

\(=3^n\cdot10-2^{n-1}\cdot10\)

\(=10\left(3^n-2^{n-1}\right)⋮10\forall n\in Z^+\)

Answer 2

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

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

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

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

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

=10.(3\(^n\)-2\(^n\).\(\dfrac{1}{2}\)) \(⋮\)10 (đpcm)

Chúc bn học tốt nha!!!!!!!