
    Design an algorithm for a learning management system that identifies the most effective learning path for a student by analyzing their assessment scores. The system needs to find the maximum achievable score progression by selecting a sequence of improving scores from the beginning of their academic record up to their current checkpoint, while also considering a specific future assessment milestone. The selected scores must form an increasing sequence to ensure the student's consistent progress, and the future milestone score must be incorporated into the final calculation. This analysis will help determine the optimal path for personalized study material recommendations based on the student's demonstrated learning pattern.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for a learning management system that identifies the most effective learning path for a student by analyzing their assessment scores. The system needs to find the maximum achievable score progression by selecting a sequence of improving scores from the beginning of their academic record up to their current checkpoint, while also considering a specific future assessment milestone. The selected scores must form an increasing sequence to ensure the student's consistent progress, and the future milestone score must be incorporated into the final calculation. This analysis will help determine the optimal path for personalized study material recommendations based on the student's demonstrated learning pattern.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for a learning management system that identifies the most effective learning path for a student by analyzing their assessment scores. The system needs to find the maximum achievable score progression by selecting a sequence of improving scores from the beginning of their academic record up to their current checkpoint, while also considering a specific future assessment milestone. The selected scores must form an increasing sequence to ensure the student's consistent progress, and the future milestone score must be incorporated into the final calculation. This analysis will help determine the optimal path for personalized study material recommendations based on the student's demonstrated learning pattern.
    

    """
    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]