The sequence of powers of two is $s_k = 2^k$, where $s_0 = 1$, $s_1 = 2$, $s_2 = 4$, and so on. To identify which of the options provides a recursive definition for powers of two, let's break them down:

Option 1:

Base case: $s_0 = 1$
Constructor: $s_k = s_{k-1} + k$ for $k \in \mathbb{Z}^+$

This is not correct because the recursive formula adds $k$ to the previous term. This does not produce powers of two. For example:

• $s_0 = 1$
• $s_1 = s_0 + 1 = 1 + 1 = 2$ (which is correct so far)
• $s_2 = s_1 + 2 = 2 + 2 = 4$ (still correct)
• $s_3 = s_2 + 3 = 4 + 3 = 7$ (not a power of two).

Option 2:

Base case: $s_0 = 1$
Constructor: $s_k = s_k + s_k$

This is incorrect because this formula suggests that $s_k = 2s_k$, which would lead to an infinite loop and is not a valid recursive definition.

Option 3:

Base case: $s_0 = 1$
Constructor: $s_k = s_{k-1} + s_{k-1} = 2s_{k-1}$ for $k \in \mathbb{Z}^+$

This is correct. The recursive step doubles the previous term, which is exactly how powers of two are generated:

• $s_0 = 1$
• $s_1 = 2 \times s_0 = 2 \times 1 = 2$
• $s_2 = 2 \times s_1 = 2 \times 2 = 4$
• $s_3 = 2 \times s_2 = 2 \times 4 = 8$, etc.

Option 4:

Base case: $s_0 = 1$
Constructor: $s_k = k \cdot s_{k-1}$

This is incorrect because multiplying by $k$ does not generate powers of two. For example:

• $s_1 = 1 \times s_0 = 1 \times 1 = 1$
• $s_2 = 2 \times s_1 = 2 \times 1 = 2$ (this is correct for now)
• $s_3 = 3 \times s_2 = 3 \times 2 = 6$ (not a power of two).

Conclusion:

The third option gives the correct recursive definition of the sequence of powers of two.

Would you like more details or have any questions?

Here are 5 follow-up questions:

1. How can we derive a closed-form expression from this recursive definition?
2. What are some other sequences that can be defined recursively?
3. How do powers of two relate to binary numbers?
4. What are some common uses of powers of two in computer science?
5. How would you modify this recursive formula to generate powers of three?

Tip: Recursive definitions are often useful in programming for algorithms like Fibonacci sequences or factorials!