题1:A=(x+2)(x-3)(x+4)(x-5)+49
=x^4-2x^3-25x^2+26x+169
=(x^2 - x - 13)^2(这一步用待定系数,设A = (ax^2 + bx + c)^2,利用两边系数相等,解方程组即可)
故A为一个完全平方数,得证.
题2:对于通项1/(n(n+1)(n+2))可化为 1/2 ×[1/(n(n+1))-1/((n+1)(n+2))].
所以原式 = 1/2 × [1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+...+1/(n(n+1))-1/((n+1)(n+2)]
= 1/2 × [1/2 - 1/((n+1)(n+2)]
< 1/2 × 1/2 = 0.25,得证.