
    Design an algorithm for an automated grading system that analyzes a student's academic progression by finding the optimal learning path. The system should identify the maximum total score achievable from a sequence of improving assessment scores up to a current checkpoint, while also considering a specific future assessment milestone. The learning path must show consistent improvement (each selected score must be higher than the previous one) and must include both scores up to the current checkpoint and the designated future assessment score. The goal is to help educators identify the most promising learning trajectory and set realistic improvement targets for students.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for an automated grading system that analyzes a student's academic progression by finding the optimal learning path. The system should identify the maximum total score achievable from a sequence of improving assessment scores up to a current checkpoint, while also considering a specific future assessment milestone. The learning path must show consistent improvement (each selected score must be higher than the previous one) and must include both scores up to the current checkpoint and the designated future assessment score. The goal is to help educators identify the most promising learning trajectory and set realistic improvement targets for students.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for an automated grading system that analyzes a student's academic progression by finding the optimal learning path. The system should identify the maximum total score achievable from a sequence of improving assessment scores up to a current checkpoint, while also considering a specific future assessment milestone. The learning path must show consistent improvement (each selected score must be higher than the previous one) and must include both scores up to the current checkpoint and the designated future assessment score. The goal is to help educators identify the most promising learning trajectory and set realistic improvement targets for students.
    

    """
    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]