Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $\left(1+\dfrac{3}{1}\right)\left(1+\dfrac{5}{4}\right)\left(1+\dfrac{7}{9}\right)\ldots\left(1+\dfrac{(2n+1)}{n^2}\right) = (n+1)^2$

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 $(k+2)^2$. …

Telescoping-product induction. First simplify the general factor $1+\frac{2r+1}{r^2}$ to $\frac{(r+1)^2}{r^2}$ using the perfect-square numerator, then multip …

• Not simplifying the general factor to $\dfrac{(r+1)^2}{r^2}$ first — attempting the induction on the un-simplified fraction $1+\frac{2r+1}{r^2}$ makes the cancellation far harder to see.
• Adding the next term instead of multiplying it (confusing this product-type induction with a sum-type induction). …