
    Design a fraud detection algorithm that analyzes a customer's transaction history to identify potentially suspicious activity. The algorithm should find the highest possible sum of legitimate increasing transactions up to a specific point in the customer's history, while also considering a future suspicious transaction. Given a sequence of transaction amounts, the algorithm must determine if the suspicious future transaction follows the customer's established spending pattern by calculating the maximum sum of transactions that form an increasing sequence up to the current point and includes the future transaction. The solution should help identify whether the future transaction represents a significant deviation from the customer's normal spending behavior. The algorithm must maintain the increasing nature of the transaction sequence to reflect natural spending progression.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a fraud detection algorithm that analyzes a customer's transaction history to identify potentially suspicious activity. The algorithm should find the highest possible sum of legitimate increasing transactions up to a specific point in the customer's history, while also considering a future suspicious transaction. Given a sequence of transaction amounts, the algorithm must determine if the suspicious future transaction follows the customer's established spending pattern by calculating the maximum sum of transactions that form an increasing sequence up to the current point and includes the future transaction. The solution should help identify whether the future transaction represents a significant deviation from the customer's normal spending behavior. The algorithm must maintain the increasing nature of the transaction sequence to reflect natural spending progression.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a fraud detection algorithm that analyzes a customer's transaction history to identify potentially suspicious activity. The algorithm should find the highest possible sum of legitimate increasing transactions up to a specific point in the customer's history, while also considering a future suspicious transaction. Given a sequence of transaction amounts, the algorithm must determine if the suspicious future transaction follows the customer's established spending pattern by calculating the maximum sum of transactions that form an increasing sequence up to the current point and includes the future transaction. The solution should help identify whether the future transaction represents a significant deviation from the customer's normal spending behavior. The algorithm must maintain the increasing nature of the transaction sequence to reflect natural spending 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]