
    In an adaptive learning system, design an algorithm to determine the most effective learning path that maximizes a student's cumulative achievement. The path should consider the student's scores from the beginning of the curriculum up to their current checkpoint, but must also account for a specific future assessment milestone. The selected sequence of modules must show progressively increasing scores to ensure proper skill building. Calculate the highest possible total score achievable through this optimized learning path while ensuring the future milestone assessment is included in the sequence.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    In an adaptive learning system, design an algorithm to determine the most effective learning path that maximizes a student's cumulative achievement. The path should consider the student's scores from the beginning of the curriculum up to their current checkpoint, but must also account for a specific future assessment milestone. The selected sequence of modules must show progressively increasing scores to ensure proper skill building. Calculate the highest possible total score achievable through this optimized learning path while ensuring the future milestone assessment is included in the sequence.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    In an adaptive learning system, design an algorithm to determine the most effective learning path that maximizes a student's cumulative achievement. The path should consider the student's scores from the beginning of the curriculum up to their current checkpoint, but must also account for a specific future assessment milestone. The selected sequence of modules must show progressively increasing scores to ensure proper skill building. Calculate the highest possible total score achievable through this optimized learning path while ensuring the future milestone assessment is included in the sequence.
    

    """
    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]