
    Design an algorithm to determine the optimal learning path that maximizes a student's cumulative performance scores up to their current checkpoint, while ensuring they can successfully incorporate a specific advanced module in their future curriculum. The learning path must follow a progressive difficulty sequence where each selected module's performance score is higher than the previous one. The algorithm should consider all completed modules up to the current checkpoint and must include the designated advanced module that comes after the checkpoint. The goal is to identify the sequence of modules that yields the highest total performance score while maintaining this increasing difficulty progression.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm to determine the optimal learning path that maximizes a student's cumulative performance scores up to their current checkpoint, while ensuring they can successfully incorporate a specific advanced module in their future curriculum. The learning path must follow a progressive difficulty sequence where each selected module's performance score is higher than the previous one. The algorithm should consider all completed modules up to the current checkpoint and must include the designated advanced module that comes after the checkpoint. The goal is to identify the sequence of modules that yields the highest total performance score while maintaining this increasing difficulty progression.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm to determine the optimal learning path that maximizes a student's cumulative performance scores up to their current checkpoint, while ensuring they can successfully incorporate a specific advanced module in their future curriculum. The learning path must follow a progressive difficulty sequence where each selected module's performance score is higher than the previous one. The algorithm should consider all completed modules up to the current checkpoint and must include the designated advanced module that comes after the checkpoint. The goal is to identify the sequence of modules that yields the highest total performance score while maintaining this increasing difficulty progression.
    

    """
    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]