Change of variable for products

Question

For this question, I want to confirm whether I'm doing it right. Here is what I have. Can anyone please help me out?

Rewrite the following expression as a single product.

Let j = k-1

$$\Biggl(3\prod_{k=0}^{n-1}\frac{k+1}{k+2}\Biggl)*\Biggl(5\prod_{k=1}^{n}\frac{k+1}{k+2}\Biggl)$$

$$ = \Biggl(3\prod_{k=0}^{n-1}\frac{k+1}{k+2}\Biggl)*\Biggl(5\prod_{j=0}^{n-1}\frac{j+2}{j+3}\Biggl)$$

$$ = \Biggl(3\prod_{k=0}^{n-1}\frac{k+1}{k+2}\Biggl)*\Biggl(5\prod_{k=0}^{n-1}\frac{k+2}{k+3}\Biggl)$$

$$ = \Biggl(15\prod_{k=0}^{n-1}\frac{k+1}{k+2}\frac{k+2}{k+3}\Biggl)$$

Answer 1

Continue: that's$$15\prod_k\frac{k+1}{k+3}=15\frac{n!}{(n+2)!/2}=\frac{30}{(n+1)(n+2)}.$$

Answer 2

A little simpler is to recognize that each product telescopes: $$ \left(3\prod_{k=0}^{n-1}\frac{k+1}{k+2}\right)\left(5\prod_{k=1}^{n}\frac{k+1}{k+2}\right) =\left(3\cdot\frac{0+1}{(n-1)+2}\right)\left(5\cdot\frac{1+1}{n+2}\right) =\frac{30}{(n+1)(n+2)} $$