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

For a positive integer $r$, the general factor is

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 product up to $r=k+1$ equals $k+2$. …

Telescoping-product induction. Multiply the assumed product $k+1$ by the next factor $1+\frac{1}{k+1}=\frac{k …

• Adding the next factor instead of multiplying (this is a product, not a sum, so the induction hypothesis must be multiplied by the $(k+1)$-th factor).
• Writing the next factor as $1+\frac1k$ again instead of the correct $1+\frac{1}{k+1}$. …