
    Design a fraud detection algorithm that analyzes transaction sequences to identify the highest-risk pattern of increasing transaction amounts leading up to a specific checkpoint transaction, while also considering a mandatory future transaction for comprehensive risk assessment. The algorithm must evaluate all possible subsequences of increasing transaction values from the start of the monitoring period up to the checkpoint transaction. The pattern must include both the transactions before the checkpoint and a specific future transaction that occurs after the checkpoint. The goal is to find the maximum sum of transaction amounts that follow this pattern, which could indicate potential money laundering or structured deposits.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a fraud detection algorithm that analyzes transaction sequences to identify the highest-risk pattern of increasing transaction amounts leading up to a specific checkpoint transaction, while also considering a mandatory future transaction for comprehensive risk assessment. The algorithm must evaluate all possible subsequences of increasing transaction values from the start of the monitoring period up to the checkpoint transaction. The pattern must include both the transactions before the checkpoint and a specific future transaction that occurs after the checkpoint. The goal is to find the maximum sum of transaction amounts that follow this pattern, which could indicate potential money laundering or structured deposits.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a fraud detection algorithm that analyzes transaction sequences to identify the highest-risk pattern of increasing transaction amounts leading up to a specific checkpoint transaction, while also considering a mandatory future transaction for comprehensive risk assessment. The algorithm must evaluate all possible subsequences of increasing transaction values from the start of the monitoring period up to the checkpoint transaction. The pattern must include both the transactions before the checkpoint and a specific future transaction that occurs after the checkpoint. The goal is to find the maximum sum of transaction amounts that follow this pattern, which could indicate potential money laundering or structured deposits.
    

    """
    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]