1*(n^2-1^2)+2*(n^2-2^2)...+n(n^2-n^2)=(1+2+..+n)*n^2-(1^3+2^3+..+n^3)其中：1+2+3+..+n=n*(n+1)/21^3+2^3+...+n^3=[n(n+1)/2]^2 所以：1*(n^2-1^2)+2*(n^2-2^2)...+n(n^2-n^2)=(1+2+..+n)*n^2-(1^3+2^3+..+n^3)＝n^3*(n+1)/2 -[n(n+1)/2]^2 =n*(n+1)(2n^2-n^2-n)/4=(n^2+n)(n^2-n)/4 =(n^4-n^2)/4 对比an^4+bn^2+c a=1/4,b=-1/4,c=0所以存在常数a、b、c,使等式1*(n^2-1^2)+2*(n^2-2^2)...+n(n^2-n^2)=an^4+bn^2+c对一切正整数n都成立.补充：1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2(n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2]=(2n^2+2n+1)(2n+1)=4n^3+6n^2+4n+12^4-1^4=4*1^3+6*1^2+4*1+13^4-2^4=4*2^3+6*2^2+4*2+14^4-3^4=4*3^3+6*3^2+4*3+1.(n+1)^4-n^4=4*n^3+6*n^2+4*n+1各式相加有(n+1)^4-1=4*(1^3+2^3+3^3...+n^3)+6*(1^2+2^2+...+n^2)+4*(1+2+3+...+n)+n4*(1^3+2^3+3^3+...+n^3)=(n+1)^4-1+6*[n(n+1)(2n+1)/6]+4*[(1+n)n/2]+n=[n(n+1)]^21^3+2^3+...+n^3=[n(n+1)/2]^2