------
## 最终结论
```python
def max_sum_increasing_subseq(arr, n, i, k):
    # Calculate the maximum sum of increasing subsequence up to index i
    max_sum = [0] * (i + 1)
    
    for j in range(i + 1):
        max_sum[j] = arr[j]
        for m in range(j):
            if arr[m] < arr[j]:
                max_sum[j] = max(max_sum[j], max_sum[m] + arr[j])
    
    # Now we need to include the kth element
    kth_element = arr[k]
    max_result = 0
    
    for j in range(i + 1):
        if arr[j] < kth_element:
            max_result = max(max_result, max_sum[j] + kth_element)
    
    return max_result

# Test cases
assert max_sum_increasing_subseq([1, 101, 2, 3, 100, 4, 5 ], 7, 4, 6) == 11
assert max_sum_increasing_subseq([1, 101, 2, 3, 100, 4, 5 ], 7, 2, 5) == 7
assert max_sum_increasing_subseq([11, 15, 19, 21, 26, 28, 31], 7, 2, 4) == 71
```