Simplify two product notations

Question

I am having trouble simplifying these two product notations.

$$\prod_{k=1}^n (k+1) \prod_{j=2}^{n-1} 1/j$$

Answer 1

See that

$$\prod_{k=1}^n(k+1)=\prod_{k=2}^{n+1}k$$

$$\prod_{k=1}^n(k+1)\prod_{j=2}^{n-1}\frac1j=\prod_{k=2}^{n+1}k\prod_{j=2}^{n-1}\frac1j$$

$$=\frac{\color{blue}{2\times3\times4\times\dots\times(n-1)}\times(n)\times(n+1)}{\color{blue}{2\times3\times4\times\dots\times(n-1)}}$$

$$\prod_{k=1}^n(k+1)\prod_{j=2}^{n-1}\frac1j=n(n+1)$$

Answer 2

Note that $$\prod_{j=2}^{n-1} \dfrac{1}{j} = \dfrac{1}{2} \dfrac{1}{3} \dots \dfrac{1}{n-1} = \dfrac{1}{(n-1)!}$$ and $$\prod_{k=1}^n (k+1) = (1+1)(2+1)\dots (n+1) = (n+1)!$$

Then $$\prod_{k=1}^n (k+1) \prod_{j=2}^{n-1} \dfrac{1}{j} = \prod_{k=1}^n (k+1) \dfrac{1}{(n-1)!} = (n+1)!\dfrac{1}{(n-1)!} = (n+1)n $$