Let $N_t$ be a $PP(3)$. Let $T_n$ be the time of the nth arrival. Find $E(N_5|N_2 = 5)$

Question

Let $N_t$ be a $PP(3)$. Let $T_n$ be the time of the nth arrival. Find:

1. $\Bbb E\left(N_5\mid N_2 = 5\right)$
2. $\Bbb E\left(T_9\mid N_2 = 5\right)$

My effort:

1. $\Bbb E\left(N_5\mid N_2 = 5\right)=\Bbb E(N_3)+5=\Bbb E(N_1+N_1+N_1)+5=1/3+1/3+1/3+5$ and
2. $\Bbb E\left(T_9\mid N_2 = 5\right)=\Bbb E(T_4+5)=4\times 1/3 +5?$

Is this correct?