Question:
Prove that $n(n+1)(n+5)$ is always divisible by 3 using mathematical induction.
Well it is quite obvious that P(1) is true. However, my question is if: $$3\lambda\frac{(k+2)(k+6)}{k(k+5)}$$ Is enough of a condition to prove that it's divisible by three. Or would I have to do this question by opening the brackets and expanding?
On
Your inductive assumption is that $3|n(n+1)(n+5)$. Since 3 is prime, that means that $3|n$ or $3|(n+1)$ or $3|(n+2)$.
You with to prove that $3|(n+1)(n+2)(n+6)$. Which is equivalent to $3|(n+1)$ or $3|(n+2)$ or $3|(n+6)$. Show that in each of the 3 cases of the inductive hypothesis, that at least one of the to-prove cases is true.
It is not enough. Note that an integer is divisible by 3 if it is of the form $3m$ where $m$ is an integer. You need to guarantee that the rational expression appeared in your question must be an integer. After that we are done.