Question

(200+1)(200-2)(200+3)(200-4)...(200+2007)(200-2008)=

 

Answer 1

$(200 + 1)(200 - 2)(200 + 3)(200 - 4)...(200+2007)(200 - 2008) =\\ prodlimits_{n = 1}^{2008} (200 + (-1)^{n-1}n)\\\ a_1 = 200 + (-1)^{0}*1 = 200 -1\\...\\ a_{200} = 200 + (-1)^{199}*200 = 200 - 200 = 0\\ ...\\ a_{2008} = 200 + (-1)^{2007}*2008 = 200 - 2008\\\ prodlimits_{n = 1}^{2008} 200 + (-1)^{n-1}n = a_1*a_2*...*a_{200}*...*a_{2008} =\\ a_1*a_2*...*0*...*a_{2008} = 0$