
    Design an investment portfolio optimization algorithm that identifies the most profitable combination of investments up to a specific evaluation period while mandating the inclusion of a specific future investment opportunity. The algorithm should analyze historical monthly returns of various investment options and find the sequence of investments with increasing returns that maximizes the total portfolio return. The selected investments must maintain a strictly increasing return rate pattern to align with the client's progressive risk tolerance strategy. The sequence must include investments only up to the current evaluation period plus exactly one specific future investment opportunity that occurs later in the timeline. The goal is to maximize the total return percentage while adhering to these sequential and timing constraints.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an investment portfolio optimization algorithm that identifies the most profitable combination of investments up to a specific evaluation period while mandating the inclusion of a specific future investment opportunity. The algorithm should analyze historical monthly returns of various investment options and find the sequence of investments with increasing returns that maximizes the total portfolio return. The selected investments must maintain a strictly increasing return rate pattern to align with the client's progressive risk tolerance strategy. The sequence must include investments only up to the current evaluation period plus exactly one specific future investment opportunity that occurs later in the timeline. The goal is to maximize the total return percentage while adhering to these sequential and timing constraints.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an investment portfolio optimization algorithm that identifies the most profitable combination of investments up to a specific evaluation period while mandating the inclusion of a specific future investment opportunity. The algorithm should analyze historical monthly returns of various investment options and find the sequence of investments with increasing returns that maximizes the total portfolio return. The selected investments must maintain a strictly increasing return rate pattern to align with the client's progressive risk tolerance strategy. The sequence must include investments only up to the current evaluation period plus exactly one specific future investment opportunity that occurs later in the timeline. The goal is to maximize the total return percentage while adhering to these sequential and timing constraints.
    

    """
    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]