Induction logic, two conditions

Question

If we assume p(n) and q(n) are true and use this to prove that p(n+1) and q(n+1) are true is this valid logic?

Answer 1

Yes: let $r(n)$ be the statement "$p(n)$ and $q(n)$". Now you can do induction as usual with respect to $n$ on the single statement $r(n)$.