Item 3.

If we take first differences we get \(1,2,4,8,16,\ldots\text{.}\) Wait, what? That's the sequence we started with. So taking second differences will give us the same sequence again. No matter how many times we repeat this we will always have the same sequence, which in particular means no finite number of differences will be constant. Thus this sequence is not \(\Delta^k\)-constant for any \(k\text{.}\)