求出100以内使得算式1+(1+2)+…+(1+2+…+n)的值小于1000的最大的自然数n

令an=1+2+...+n;
an=1/2[n(n+1)]=1/2n^2+1/2n;
a(n-1)=1/2(n-1)^2+1/2(n-1)
..
a1=1/2+1/2;
sn=a1+a2+..an=1/2(1+2^2+3^2+..n^2)+1/2(1+2+3+..+n)
=n(n+1)(2n+1)/12+n(n+1)/4
=n(n+1)(2n+4)/12
=n(n+1)(n+2)/6>n^3/6;
算式1+(1+2)+…+(1+2+…+n)的值小于1000;
所以n(n+1)(n+2)/6