(1)g(x)=f4(x)+2f5(x)+3f6(x)=(1+)4+2(1+)5+3(1+)6, ∴g(x)中含x2项的系数为+2+3=1+10+45=56.(3分) (2)证明:由题意,pn=2n-1.(5分) ①当n=1时,p1(a1+1)=a1+1,成立; ②假设当n=k时,pk(a1a2…ak+1)≥(1+a1)(1+a2)…(1+ak)成立, 当n=k+1时,(1+a1)(1+a2)…(1+ak)(1+ak+1)≤2k-1(a1a2…ak+1)(1+ak+1) =2k-1(a1a2…akak+1+a1a2…ak+ak+1+1).(*) ∵ak>1,a1a2…ak(ak+1-1)≥ak+1-1,即a1a2…akak+1+1≥a1a2…ak+ak+1, 代入(*)式得(1+a1)(1+a2)…(1+ak)(1+ak+1)≤2k(a1a2…akak+1+1)成立. 综合①②可知,pn(a1a2…an+1)≥(1+a1)(1+a2)…(1+an)对任意n∈N*成立.(10分) |