Question

1+4+9+16+25+…+㎡＝?

Answer 1

1+2*2+3*3+...+m*m=1+1*2+2+2*3+3+3*4+4+.(m-1)m+m

=(1+2+3+...+m)+[1*2+2*3+3*4+.(m-1)*m]

=m(m+1)/2+[1*2*3+2*3*3+3*4*3+.(m-1)*m*3]/3

=m(m+1)/2+[1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+.(m-1)m(m+1)-(m-2)(m-1)m]/3

=m(m+1)/2+(m-1)m(m+1)/3=m(m+1)(2m+1)/6