2014/050) find the integral part of (2005)^3/(2003 * 2004) – (2003)^3 / (2004 * 2005)

To keep the arithmetic simple put

2004 = x to get

(x+1)^3/(x(x-1) - (x-1)^3/(x(x+1)

= ((x+1)^4 – (x-1)^4) / x(x+1)(x-1)

= 2 (4x^3 + 4x)/(x(x+1)(x-1)

= 8x(x^2+1)/x(x^2-1)

= 8(x^2 + 1)/(x^2-1)

= 8 + 2//(x^2-1)

Now the beauty is for any x > 2 the integral part is 8

Hence ans is 8.