Question
If n is positive integer, show that 3^{n+2}-2^{n+2}+3^n-2^n is divisible by 10
If n is positive integer, show that 3^{n+2}-2^{n+2}+3^n-2^n is divisible by 10
3^{n+2}-2^{n+2}+3^n-2^n
If n=1
3^{1+2}-2^{1+2}+3^1-2^1
=27-8+3-2 = 20
which is divisible by 10
If n>1
3^{n+2}-2^{n+2}+3^n-2^n
=3^{n+2}+3^n-(2^{n+2}+2^n)
=3^n(3^2+1)-2^n(2^2+1)
=10\cdot 3^n-5\cdot 2^n
=10\cdot 3^n-10\cdot 2^{n-1}
which is divisible by 10
Therefore
3^{n+2}-2^{n+2}+3^n-2^n is divisible by 10 for any positive integer n