
    In a patient monitoring system, develop an algorithm to identify the most concerning pattern of deteriorating vital signs that could indicate increasing health risks. The algorithm should analyze the patient's historical vital sign measurements up to their current visit, considering only sequences where the readings show a progressive increase (potentially indicating worsening condition). The analysis must incorporate a specific future scheduled check-up measurement to assess potential trajectory of the condition. The goal is to quantify the maximum severity score by summing the selected vital sign measurements that form an increasing sequence up to the current visit, while mandatorily including the projected measurement from the scheduled future check-up.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    In a patient monitoring system, develop an algorithm to identify the most concerning pattern of deteriorating vital signs that could indicate increasing health risks. The algorithm should analyze the patient's historical vital sign measurements up to their current visit, considering only sequences where the readings show a progressive increase (potentially indicating worsening condition). The analysis must incorporate a specific future scheduled check-up measurement to assess potential trajectory of the condition. The goal is to quantify the maximum severity score by summing the selected vital sign measurements that form an increasing sequence up to the current visit, while mandatorily including the projected measurement from the scheduled future check-up.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    In a patient monitoring system, develop an algorithm to identify the most concerning pattern of deteriorating vital signs that could indicate increasing health risks. The algorithm should analyze the patient's historical vital sign measurements up to their current visit, considering only sequences where the readings show a progressive increase (potentially indicating worsening condition). The analysis must incorporate a specific future scheduled check-up measurement to assess potential trajectory of the condition. The goal is to quantify the maximum severity score by summing the selected vital sign measurements that form an increasing sequence up to the current visit, while mandatorily including the projected measurement from the scheduled future check-up.
    

    """
    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]