$6\mathrm{.}2.$ You are going on a long trip. You start on the road at mile post $0.$ Along the way there are $n$ hotels, at mile posts $a_i{a}_{n}x(200-x{)}^2{a}_1,$ where each $a_{i}is$ measured from the starting point. The only places you are allowed $to$ stop are $at$ these hotels, but you can choose which $of$ the hotels you stop $at.$ You must stop $at$ the final hotel $(at$ distance $a_n),$ which $is$ your destination.

You'd ideally like $to$ travel $200$ miles a day, but this may not $be$ possible $(dependingon$ the spacing $of$ the hotels$).If$ you travel $x$ miles during a day, the penalty for that day $is(200-x{)}^2.$ You want $to$ plan your trip $soasto$ minimize the total penalty$-$that $is,$ the sum, over all travel days, $of$ the daily penalties.

Give $an$ efficient algorithm that determines the optimal sequence $of$ hotels $at$ which $to$ stop.