Normally the usual format of the principle of the mathematical induction is given by $P(k) \implies P(k+1)$, but I can't help notice that there are some people using the other notation $P(k-1) \implies P(k)$ in their proof; is there any essential difference between these two notations? Which notation is better when solving the induction problem?
P.S. I notice that sometimes using $P(k-1) \implies P(k)$ seems to make the problem easier to solve; is this my illusion?
On
Both can be used ... but you'll have to pay attention to what your base cases are and what that means for your step.
For example, if you are trying to prove that something is true for all numbers $2$ and up, then you obviously pick $2$ as your base case, but if you then use $P(n)\to P(n+1)$ as your step, you have to assume that $n \geq 2$, while if you use $P(n-1)\to P(n)$ you assume that $n \geq 3$
But yes, sometimes it is just a little easier to work with / think about $P(n-1)\to P(n)$ rather than $P(n)\to P(n+1)$
Essentially, they are the same. In some applications, the expression for $k$ might be easier to manipulate (into an expression in terms of $k-1$) than the expression for $k+1$ (into an expression in terms of $k$). Or for some structural arguments, it might be easier to make observations from a smaller structure ($k-1$). But these can change from person to person and I don't think using $P(k-1) \to P(k)$ makes too much difference. So using $P(k) \to P(k+1)$ or $P(k-1) \to P(k)$ is more about style.