Solution: Prove 1^3+2^3+3^3+...+n^3 = (1+2+...+n)^2.....(1)
this we can prove by induction method
case 1. let n=1
1^3=1²
1=1
case 2. for n=1 this is true
case 3.now assume for n-1 it is true
so, (1+2+3+...........+n-1)²=1^3+2^3+3^3+...........(n-1)^3
then to prove it for n e can expand equation (1) as ,
(1+2+3+.......+n-1)²+2n(1+2+3+..........+n-1)+n²=1³+2³+3³+......+(n-1)³.....+n³
from case 1 and 2 we get
n²+2n(1+2+3+.........(n-1)=n³
there for
By mathematical induction it is prove that
1^3+2^3+3^3+...+n^3 = (1+2+...+n)^2