Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $1.2.3 + 2.3.4 + \ldots + n(n+1)(n+2) = \dfrac{n(n+1)(n+2)(n+3)}{4}$

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 $\dfrac{(k+1)(k+2)(k+3)(k+4)}{4}$.

Adding the next term $(k+1)(k+2)(k+3)$ to both sides of the hypothesis: …

Product-term induction. Add the next term $(k+1)(k+2)(k+3)$ to the closed form, factor out the common product $(k+1)(k+2)(k+3)$, and …

• Misidentifying the next term as $(k+1)(k+2)(k+3)(k+4)$ instead of the correct $(k+1)(k+2)(k+3)$ (matching the pattern $n(n+1)(n+2)$ at $n=k+1$). …