Don't shoot the animal, hunter!

The animal escapes if the hunter misses at every distance from $2A$ to $1729A$ . The probability that this happens is

$\displaystyle\prod_{n=2}^{1729} (1 - \frac{A^{2}}{(nA)^{2}}) = \displaystyle\prod_{n=2}^{1729} (1 - \frac{1}{n^{2}}) = \displaystyle\prod_{n=2}^{1729} \frac{(n - 1)(n + 1)}{n*n}$ .

Now this a telescoping product, since if we write all the successive terms in this fashion we end up canceling all the middle terms and end up with an answer of just

$(\dfrac{1}{2})(\dfrac{1730}{1729}) = \dfrac{865}{1729}$ .

This means that $a = 865, b = 1729$ and $b - a = \boxed{864}$ .

(To demonstrate how the telescoping works, I'll write out the product from $n = 2$ to $n = 5$ :

$(\dfrac{1*3}{2*2})(\dfrac{2*4}{3*3})(\dfrac{3*5}{4*4})(\dfrac{4*6}{5*5}) = (\dfrac{1}{2})(\dfrac{6}{5})$ .)

good problem,becomes simple once some one sees it as a telescopic series,, but what would have made this problem really easy was if the hunter had infinite chances to hunt the deer,., in that case ,, the probability would have been (1/2) and the answer 1