已知点O(0,0),Ao(0,1),An(6,7),点A1,A2,……,An-1(nN,n＞=2)是线段AoAn的n等分点,则|向量OAo+OA1+……+OAn-1+OAn|等于多少?

Question

已知点O(0,0),Ao(0,1),An(6,7),点A1,A2,……,An-1(nN,n＞=2)是线段AoAn的n等分点,则|向量OAo+OA1+……+OAn-1+OAn|等于多少?

Answer 1

向量OAo+OA1+……+OAn-1+OAn=(OAn-AoAn)+(OAn-A1An)+(OAn-A2An)+(OAn-A3An)+...+(OAn-An-1An)
=(n+1)OAn-(AoAn+A1An+A2An...+An-1An),右边画个图就可以明了,An-1An=(AoAn)/n
原式=(n+1)OAn-AoAn{[(n-1)/n]+[(n-2)/n]+[(n-3)/n]+[(n-4)/n]+...+1/n}
=(n+1)OAn-AoAn{(n-1)+(n-2)+n-3)+(n-4)+...+1}/n
=(n+1)OAn-AoAn*[1+(n-1)*(n-1)]/(2n) （这是求和公式）
=(n+1)OAn-AoAn*(n-1)/2=(n+1)(6,7)-[(n-1)/2]*(6,6)=(6n+6,7n+7)-(3n-3,3n-3)=(3n+9,4n+10)