Simplify Product $\prod_{k=2}^{n} \left(1 - \frac{2}{k (k+1)}\right)$

Question

I try to simplify an expression (wolfram link) but i really do not know enough about the topic.

$\prod_{k=2}^{n} \left(1 - \frac{2}{k (k+1)}\right)$

I can see the result but i do not know how to get there. I promised a friend to help with homework but now we are both stuck. Can someone show us how to think about a problem like that.

The result is supposed to be $\frac{(n+2)}{(3 n)}$ so i assume that i can split the product in two products, remove some elements from either product and somehow elimitate the remaining products with each other. I failed miserably when i tried it though.

Answer 1

$$1-\frac2{k(k+1)}=\frac{k^2+k-2}{k(k+1)}=\frac{(k-1)(k+2)}{k(k+1)}=\frac{\frac{k-1}{k+1}}{\frac k{k+2}}=\frac{v_k}{v_{k+1}}$$ where $$v_k=\frac{k-1}{k+1}$$ so the desired product gives by telescoping $$\frac{v_2}{v_{n+1}}=\frac{n+2}{3n}$$

Answer 2

Hint: $$ 1 - \frac 2{k(k+1)} =\frac {k^2 + k - 2}{k(k+1)} =\frac {(k-1)(k+2)}{k(k+1)} $$

Then when you make the product of these, most terms telescope.