Answer:
The answer is True
Step-by-step explanation:
A mathematical induction consists in only 2 steps:
First step: Show the proposition is true for the first one valid integer number.
Second step: Show that if any one is true then the next one is true
Finally, if first step and second step are true, then the complete proposition is true.
So, given [tex]2*n^3+3*n^2+n[/tex]
First step: using and replacing n=2 (the first valid integer number >1)
[tex]2*(2)^3 +3*(2)^2+2=30[/tex]
[tex]\frac{30}{6} =5[/tex]
As the result is an integer number, so the first step is true.
Second step: using any next number, [tex]n+1[/tex], let it replace
[tex]2*(n+1)^3+3*(n+1)^2+(n+1)\\2*(n^3+3*n^2+3*n+1)+3*(n^2+2*n+1)^2+(n+1)\\2*n^3+6*n^2+6*n+2+3*n^2+6*n+3+n+1\\(2*n^3+3*n^2+n)+(6*n^2+12*n+6)[/tex]
As the First step is true, we know that
[tex]2*n^3+3*n^2+n[/tex][tex]=6*k[/tex],
So let it replace in the previous expression
[tex]6*k+6*(n^2+2*n+1)\\6*[k+(n^2+2*n+1)][/tex]
Finally
[tex]\frac{6*[k+(n^2+2*n+1)]}{6} =k+(n^2+2*n+1)[/tex]
where the last expression is an integer number
So the second step is true, and the complete proposition is True
