Yes, the algorithm you described is called "Binary Search." Here is a Python implementation of the binary search algorithm:

```python
def binary_search(arr, target, start, end):
    if start > end:
        return -1
    
    mid = (start + end) // 2
    
    if arr[mid] == target:
        return mid
    elif arr[mid] > target:
        return binary_search(arr, target, start, mid - 1)
    else:
        return binary_search(arr, target, mid + 1, end)

# Example usage
arr = [1, 3, 5, 7, 9, 11, 13, 15]
target = 7
result = binary_search(arr, target, 0, len(arr) - 1)

if result != -1:
    print(f"Target found at index: {result}")
else:
    print("Target not found in the list.")
```

You can adjust the `arr` and `target` values to test the algorithm with different inputs.