POJ1759

    同樣是二分搜索，這道題也挺好的，值得做做。首先你需要對公式Hi = (Hi-1 + Hi+1)/2 - 1做一個變換，改成Hi+1 = 2*Hi - Hi-1 + 2。使用這樣的遞推公式，那麼只要知道H2就可以把剩下的都求出來，那麼如何這道題就明朗多了。首先枚舉H2的值，然後遞推出剩餘的H值，然後判斷是否滿足Hi >= 0, for all 1 <= i <= N的要求。當我們用H1和H2來表示Hi時，我們可以發現一個規律，就是Hi = (i-2) * H2 - (i-1) * H1 + (i-1)*(i-2)，也就意味着H2和Hn成正相關，求最小的Hn也就變成了求最小的H2。
    順便說一下，有浮點數的時候精度有可能是一個要注意的問題，如果你覺得方法已經對了，那你就想想如何處理精度問題，倒不一定是越精確越好，因爲要AC重要的是要和標答一致，所以有時就有點坑了。

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代碼（G++）：

#include <iostream>
#include <iomanip>
#include <cmath>

#define MAXN 1003
using namespace std;

int n;
double h[MAXN], ans;

bool is_ok(double h2)
{
    if(h2 < 0) return false;
    h[2] = h2;
    for(int i=3; i<=n; i++)
    {
        h[i] = h[i-1]*2 - h[i-2] + 2;
        if(h[i] < 0) return false;
    }
    ans = h[n];
    return true;
}

int main()
{
    double lb, ub, mid;
    while(cin>>n>>h[1])
    {
        lb = -1;
        ub = h[1];
        for(int i=0; i<100; i++)
        {
            mid = (lb + ub) / 2;
            if(is_ok(mid)) ub = mid;
            else lb = mid;
        }
        //ans = abs((n-1)*ub - (n-2)*h[1] + (n-1)*(n-2));
        //ans = floor(ans * 100) / 100;
        cout<<setiosflags(ios::fixed)<<setprecision(2)<<ans<<endl;
    }
    return 0;
}

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題目（注意：題幹中的圖沒有貼過來）：
Garland
Time Limit: 1000MS Memory Limit: 10000K

Description

The New Year garland consists of N lamps attached to a common wire that hangs down on the ends to which outermost lamps are affixed. The wire sags under the weight of lamp in a particular way: each lamp is hanging at the height that is 1 millimeter lower than the average height of the two adjacent lamps. 

The leftmost lamp in hanging at the height of A millimeters above the ground. You have to determine the lowest height B of the rightmost lamp so that no lamp in the garland lies on the ground though some of them may touch the ground. 

You shall neglect the lamp's size in this problem. By numbering the lamps with integers from 1 to N and denoting the ith lamp height in millimeters as Hi we derive the following equations: 

H1 = A 
Hi = (Hi-1 + Hi+1)/2 - 1, for all 1 < i < N 
HN = B 
Hi >= 0, for all 1 <= i <= N 

The sample garland with 8 lamps that is shown on the picture has A = 15 and B = 9.75. 
Input

The input file consists of a single line with two numbers N and A separated by a space. N (3 <= N <= 1000) is an integer representing the number of lamps in the garland, A (10 <= A <= 1000) is a real number representing the height of the leftmost lamp above the ground in millimeters.
Output

Write to the output file the single real number B accurate to two digits to the right of the decimal point representing the lowest possible height of the rightmost lamp.
Sample Input

692 532.81
Sample Output

446113.34