
    Design an algorithm for an adaptive learning system that determines the most effective learning path by analyzing student performance scores. The system needs to identify the maximum sum of improving scores from the beginning of the course up to a current checkpoint, while also considering performance in a specific future milestone module. The sequence of scores must be strictly increasing to ensure the student is progressively improving their understanding. The algorithm should help determine the optimal difficulty progression that maintains student engagement while preparing them for the upcoming milestone module.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for an adaptive learning system that determines the most effective learning path by analyzing student performance scores. The system needs to identify the maximum sum of improving scores from the beginning of the course up to a current checkpoint, while also considering performance in a specific future milestone module. The sequence of scores must be strictly increasing to ensure the student is progressively improving their understanding. The algorithm should help determine the optimal difficulty progression that maintains student engagement while preparing them for the upcoming milestone module.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for an adaptive learning system that determines the most effective learning path by analyzing student performance scores. The system needs to identify the maximum sum of improving scores from the beginning of the course up to a current checkpoint, while also considering performance in a specific future milestone module. The sequence of scores must be strictly increasing to ensure the student is progressively improving their understanding. The algorithm should help determine the optimal difficulty progression that maintains student engagement while preparing them for the upcoming milestone module.
    

    """
    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]