Calculate $((2^3+1)/(2^3-1))((3^3+1)/(3^3-1))...((10^3+1)/(10^3-1))$

Question

Calculate $$\frac{(2^3+1)}{(2^3-1)}\cdot\frac{(3^3+1)}{(3^3-1)}\cdots \frac{(10^3+1)}{(10^3-1)}.$$

Answer 1

Hint. More generally, for any integer $n>1$, $$\prod_{k=2}^{n}\frac{k^3+1}{k^3-1}=\prod_{k=2}^{n}\frac{(k+1)((k-1)k+1)}{(k-1)(k(k+1)+1)}=\prod_{k=2}^{n}\frac{k+1}{k-1}\cdot \prod_{k=2}^{n}\frac{(k-1)k+1}{k(k+1)+1}.$$

Answer 2

Hint: $$x^3+1=(x+1)(x^2-x+1)=(x+1)(x(x-1)+1)$$ while $$(x-1)^3-1=((x-1)-1)((x-1)^2+(x-1)+1)=(x-2)(x(x-1)+1)$$

Answer 3

Hint: $$(x^3+1)/(x^3-1)=(x+1)\times(x^2+1-x)\times(x-1)^{-1}\times(x^2+1+x)^{-1}$$
replacing $x$ by $x+1$ and multiplying, the denominator of one one gets cancelled with numerator of other, similarly for every two consecutive terms .
For eg, $$8^2+1+8=9^2+1-9=73$$