
    In a hospital's patient monitoring system, develop an algorithm to identify the most concerning sequence of deteriorating vital signs up to the current monitoring point while accounting for a critical future measurement. The sequence must show progressively worsening values (increasing numbers indicating deterioration) from the start of monitoring up to the current time index. The algorithm must include both the optimal sequence of readings up to the current point and a specific future measurement at position k to help predict potential patient decline. The goal is to find the sequence with the highest sum of measurements that could indicate the most severe pattern of deterioration. This analysis will help healthcare providers identify high-risk patterns and take preventive measures before critical deterioration occurs.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    In a hospital's patient monitoring system, develop an algorithm to identify the most concerning sequence of deteriorating vital signs up to the current monitoring point while accounting for a critical future measurement. The sequence must show progressively worsening values (increasing numbers indicating deterioration) from the start of monitoring up to the current time index. The algorithm must include both the optimal sequence of readings up to the current point and a specific future measurement at position k to help predict potential patient decline. The goal is to find the sequence with the highest sum of measurements that could indicate the most severe pattern of deterioration. This analysis will help healthcare providers identify high-risk patterns and take preventive measures before critical deterioration occurs.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    In a hospital's patient monitoring system, develop an algorithm to identify the most concerning sequence of deteriorating vital signs up to the current monitoring point while accounting for a critical future measurement. The sequence must show progressively worsening values (increasing numbers indicating deterioration) from the start of monitoring up to the current time index. The algorithm must include both the optimal sequence of readings up to the current point and a specific future measurement at position k to help predict potential patient decline. The goal is to find the sequence with the highest sum of measurements that could indicate the most severe pattern of deterioration. This analysis will help healthcare providers identify high-risk patterns and take preventive measures before critical deterioration occurs.
    

    """
    dp = [[0 for i in range(n)] for i in range(n)]
    for i in range(n):
        if a[i] > a[0]:
            dp[0][i] = a[i] + a[0]
        else:
            dp[0][i] = a[i]
    for i in range(1, n):
        for j in range(n):
            if a[j] > a[i] and j > i:
                if dp[i - 1][i] + a[j] > dp[i - 1][j]:
                    dp[i][j] = dp[i - 1][i] + a[j]
                else:
                    dp[i][j] = dp[i - 1][j]
            else:
                dp[i][j] = dp[i - 1][j]
    return dp[index][k]