We are asked to determine three consecutive even integers. Let those integers be:
[tex]n_1,n_2,n_3[/tex]Since the integers are consecutive we have the following relationships:
[tex]\begin{gathered} n_2=n_1+2 \\ n_3=n_1+4 \end{gathered}[/tex]We are also given that two times the second equals the sum of the first and third. This can be expressed mathematically as:
[tex]2n_2=n_1+n_3[/tex]Now we can substitute the value of n3 from the previous relationship:
[tex]2n_2=n_1+n_1+4[/tex]Adding like terms:
[tex]2n_2=2n_1+4[/tex]We can also substitute the value of n3:
[tex]2(n_1+2)=2n_1+4[/tex]We get:
[tex]2n_1+4=2n_1+4[/tex]This is valid for any "n". Therefore any three consecutive integers are a solution.
