一个小球从h米高处自由落下,每次着地后都反弹到前一次高度的2/3,求小球经过的总路程【数列应用】
2019-05-30
一个小球从h米高处自由落下,每次着地后都反弹到前一次高度的2/3,求小球经过的总路程【数列应用】
优质解答
h+h×(2/3)+h×(2/3)+h×(2/3)²+h×(2/3)²+...+h×(2/3)^(n-1) +h×(2/3)^(n-1)
=2h[1+(2/3)+(2/3)²+...+(2/3)^(n-1)]-h
=2h[1-(2/3)ⁿ]/(1-2/3) -h
=6h -6h×(2/3)ⁿ-h
=5h-6h×(2/3)ⁿ
n->+∞时,6h×(2/3)ⁿ->0 5h-6h×(2/3)ⁿ->5h
lim[5h-6h×(2/3)ⁿ]=5h
小球经过的总路程为5h.
h+h×(2/3)+h×(2/3)+h×(2/3)²+h×(2/3)²+...+h×(2/3)^(n-1) +h×(2/3)^(n-1)
=2h[1+(2/3)+(2/3)²+...+(2/3)^(n-1)]-h
=2h[1-(2/3)ⁿ]/(1-2/3) -h
=6h -6h×(2/3)ⁿ-h
=5h-6h×(2/3)ⁿ
n->+∞时,6h×(2/3)ⁿ->0 5h-6h×(2/3)ⁿ->5h
lim[5h-6h×(2/3)ⁿ]=5h
小球经过的总路程为5h.