Many students have asked me about Mathematical Induction, like whats the idea behind it. Especially since I always emphasize the importance of $P(1)$.

So lets try to make some sense of this long and mundane proof.

Our first step always seek to prove that $P(1)$ is true. And then $P(k)$ is our induction hypothesis, think of it as an assumption (so it might or might not be true). We then seek to prove $P(k+1)$ is true with the use of $P(k)$. Should we succeed in proving it, we establish some relationship between $P(k)$ and $P(k+1)$. But if we trace back, $P(k)$ was a mere assumption. So where do we go from here?

If we trace back to $P(1)$, then we will create a domino effect like the above. Since we have that $P(1)$ is true definitely, using the proven relationship with $P(k)$ and $P(k+1)$, we can establish $P(2)$ is true. Here, we create a huge domino effect up to n.

Inquisitive students have asked me before, what if k is greater than n? Now that’s a much profound question to be answered. But curious students can google “Strong Mathematical Induction”.