The property was proved for $n=1$ The property is correct if n is changed by $n+1$
Work Step by Step
Let's prove the property for $n=1$: $n^4-n+4=1^4-1+4=4=2(2)$ 2 is a factor. It is correct! Suppose that the property is correct, that is: $n^4-n+4$ has 2 as a factor for all integers $n\geq1$ Now, let's prove the property for $n+1$: $(n+1)^4-(n+1)+4=n^4+4n^3+6n^2+4n+1-n-1+4=(n^4-n+4)+(4n^3+6n^2+4n+1-1)=(n^4-n+4)+2(2n^3+3n^2+2n)$ It is a sum of two terms that have 2 as a factor.