Answer
(a) $P(0)$ is true.
(b) $5^k-1$ is divisible by 4.
(c) $5^{k+1}-1$ is divisible by 4.
(d) if $P(k)$ is true, then $P(k+1)$ will also be true.
Work Step by Step
(a) For $n=0$, we have $5^0-1=4$ is divisible by 4, thus $P(0)$ is true.
(b) For $n=k$, we have $P(k)$ as $5^k-1$ is divisible by 4.
(c) For $n=k+1$, we have $P(k+1)$ as $5^{k+1}-1$ is divisible by 4.
(d) In the proof as described in the exercise, we must prove that if $P(k)$ is true, then $P(k+1)$ will also be true.