Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $1.2 + 2.3 + 3.4 + \ldots + n.(n+1) = \left[\dfrac{n(n+1)(n+2)}{3}\right]$

Let $P(n)$ be the statement

Base case: For $n=1$,

So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$:

We must show the sum up to $(k+1)$ equals $\dfrac{(k+1)(k+2)(k+3)}{3}$.

Adding the next term $(k+1)(k+2)$ to both sides of the hypothesis: …

Product-term induction. Add the next term $(k+1)(k+2)$ to the closed form, factor out the common $(k+1)(k+2 …

• Misidentifying the $(k+1)$-th term as $k(k+1)$ (the $k$-th term) instead of $(k+1)(k+2)$.
• Skipping the factoring step and expanding everything out, which is much more error-prone for verifying the match with the target closed fo …