
    Design an algorithm for an automated assessment system that analyzes a student's writing performance trajectory. Given a sequence of past writing assignment scores and a specific future assignment score, determine the highest possible cumulative performance trend that can be achieved. The trend must show consistent improvement from the beginning up to any chosen current assignment, and must include the specified future assignment score. The analysis should help identify the student's best potential growth pattern while maintaining a strictly increasing sequence of scores. The goal is to find the maximum total score possible while adhering to these progression requirements.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for an automated assessment system that analyzes a student's writing performance trajectory. Given a sequence of past writing assignment scores and a specific future assignment score, determine the highest possible cumulative performance trend that can be achieved. The trend must show consistent improvement from the beginning up to any chosen current assignment, and must include the specified future assignment score. The analysis should help identify the student's best potential growth pattern while maintaining a strictly increasing sequence of scores. The goal is to find the maximum total score possible while adhering to these progression requirements.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for an automated assessment system that analyzes a student's writing performance trajectory. Given a sequence of past writing assignment scores and a specific future assignment score, determine the highest possible cumulative performance trend that can be achieved. The trend must show consistent improvement from the beginning up to any chosen current assignment, and must include the specified future assignment score. The analysis should help identify the student's best potential growth pattern while maintaining a strictly increasing sequence of scores. The goal is to find the maximum total score possible while adhering to these progression requirements.
    

    """
    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]