5 / 2 of a is equivalent to 3 / 1 of B, a: B = (): ()

5 / 2 of a is equivalent to 3 / 1 of B, a: B = (): ()

5 / 2 of a is equivalent to 3 / 1 of B
It is known that a ∶ B = 1 / 3 ∶ 2 / 5 = 5 ∶ 6

For any rational number AB, we define *: a * b = (a + 1) (B + 2) 1 / 2, find 1 * 2 + 3 * 4 + 5 * 6 +. + 2003 * 2004

A * b = 1 / (a + 1) (B + 2) so: a * (a + 1) = 1 / (a + 1) (a + 1 + 2) = 1 / (a + 1) (a + 3) = (1 / (a + 1) - 1 / (a + 3)) / 2
So:
1*2+3*4+5*6+.+2003*2004=((1/2-1/4)+(1/4-1/6)+(1/6-1/8)+.+(1/2004-1/2006))/2
=(1/2-1/2006)/2
=(2004/4012)/2
=1002/4012
=501/2006

If a # B = B − AA × B, then 1 # 2-2 # 3-3 # 4 - -2002#2003-2003#2004=______ ．

A#B=B−AA×B=BA×B-AA×B=1A-1B；1#2-2#3-3#4-… -2002#2003-2003#2004，=11-12-（12-13）-（13-14）-… -(12002-12003) - (12003-12004), = 12004; answer: if a # B = B − AA × B, then 1 # 2-2 # 3-3 # 4-l-2002 # 2003-2003 # 2004 = 12004