
    Design a system to identify the strongest positive trend in a student's academic performance that could indicate their potential for success. The system should analyze the student's performance metrics up to a current checkpoint and factor in their projected performance at a critical future assessment. Given a series of performance scores and a specified future assessment point, find the maximum sum of consistently improving scores up to the current checkpoint that also includes the projected score at the future assessment. The performance trend must show steady improvement to be considered valid, and the future assessment score must be included in the final calculation. The goal is to identify the most promising trajectory that could help predict student success and inform early intervention strategies.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a system to identify the strongest positive trend in a student's academic performance that could indicate their potential for success. The system should analyze the student's performance metrics up to a current checkpoint and factor in their projected performance at a critical future assessment. Given a series of performance scores and a specified future assessment point, find the maximum sum of consistently improving scores up to the current checkpoint that also includes the projected score at the future assessment. The performance trend must show steady improvement to be considered valid, and the future assessment score must be included in the final calculation. The goal is to identify the most promising trajectory that could help predict student success and inform early intervention strategies.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a system to identify the strongest positive trend in a student's academic performance that could indicate their potential for success. The system should analyze the student's performance metrics up to a current checkpoint and factor in their projected performance at a critical future assessment. Given a series of performance scores and a specified future assessment point, find the maximum sum of consistently improving scores up to the current checkpoint that also includes the projected score at the future assessment. The performance trend must show steady improvement to be considered valid, and the future assessment score must be included in the final calculation. The goal is to identify the most promising trajectory that could help predict student success and inform early intervention strategies.
    

    """
    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]