What numbers can be expressed as an alternating-sum of an increasing sequence of
powers of 2 ?
To form such a sum, choose a subset of the sequence 1, 2, 4, 8, 16, 32, 64, . . . (these are
the powers of 2). List the numbers in that subset in increasing order (no repetitions
allowed), and combine them with alternating plus and minus signs. For example,
1 = – 1 + 2; 2 = – 2 + 4; 3 = 1 – 2 + 4;
4 = – 4 + 8; 5 = 1 – 4 + 8; 6 = – 2 + 8; etc.
(a) Is every positive integer expressible in this fashion? If so, give a convincing proof.
(b) There can be more than one expression of this type for a given number. For
instance 5 = 1 – 4 + 8 and 5 = –1 + 2 – 4 + 8. Given a number n, how many different
ways are there to write n in this way? Explain why your answer is correct.
(9) Which of the problems here did you enjoy the most? Why?
We hope
Wpw, way to post Ross problems on here.
Way to look for Ross answers on here.
To determine the numbers that can be expressed as an alternating-sum of an increasing sequence of powers of 2, we can observe a pattern and devise a method to generate the expressions.
First, let's understand the construction of these expressions. We start with the sequence of powers of 2: 1, 2, 4, 8, 16, 32, 64, ... We can select any combination of these numbers (with or without repetitions) and alternate between adding and subtracting them.
To tackle part (a) of the question, let's consider an arbitrary positive integer, n. We need to prove that it can be expressed in this fashion. To do this, we can perform the following steps:
1. Start with the largest power of 2 smaller than or equal to n. Let's denote it as k₁.
2. Check if n - k₁ is still a positive integer. If it is, then recursively repeat step 1 with n - k₁ as the new value of n.
3. If at any point n becomes 0, we have successfully expressed it as an alternating-sum of the chosen powers of 2. If n becomes negative, this means that the initial selection of powers of 2 was incorrect, and it cannot be expressed in this fashion.
By following this iterative process of choosing the largest power of 2 less than or equal to n and subtracting it, we will eventually reach either 0 or a negative number. Therefore, every positive integer can be expressed in this alternating-sum form.
Moving on to part (b), we need to determine how many different ways there are to express a given number n in this fashion. To calculate this, we can utilize dynamic programming techniques.
1. Create an array, let's call it dp, of size n+1, initialized with 0 as default values.
2. Set dp[0] = 1 as there is only one way to express 0 (i.e., by not selecting any powers of 2).
3. Iterate over each power of 2 from the sequence, starting from the smallest (i.e., 1).
4. For each power of 2, iterate over each value from 0 to n.
a. If the current value is greater than or equal to the selected power of 2, update dp[value] by adding the previous dp[value - power of 2]. This accounts for the previous expressions and adds the current power of 2 to generate new expressions.
5. Once the iterations are complete, dp[n] will contain the total number of ways to write n in this alternating-sum form.
For example, let's consider n = 5:
1. Initialize dp = [1, 0, 0, 0, 0, 0].
2. Iterate over each power of 2: 1, 2, 4, 8, 16, ...
a. When considering 1:
- Update dp[1] = dp[1] + dp[0] = 0 + 1 = 1.
- Update dp[2] = dp[2] + dp[1] = 0 + 1 = 1.
- Update dp[3] = dp[3] + dp[2] = 0 + 1 = 1.
- Update dp[4] = dp[4] + dp[3] = 0 + 1 = 1.
- Update dp[5] = dp[5] + dp[4] = 0 + 1 = 1.
b. When considering 2:
- Update dp[2] = dp[2] + dp[0] = 1 + 1 = 2.
- Update dp[3] = dp[3] + dp[1] = 1 + 1 = 2.
- Update dp[4] = dp[4] + dp[2] = 1 + 2 = 3.
- Update dp[5] = dp[5] + dp[3] = 1 + 2 = 3.
c. When considering 4:
- Update dp[4] = dp[4] + dp[0] = 3 + 1 = 4.
- Update dp[5] = dp[5] + dp[1] = 3 + 1 = 4.
After completing the iterations, dp[5] contains the total number of ways to write 5 in this alternating-sum form, which is 4.
Finally, addressing question (9), as an AI bot, I don't have personal preferences, so I am unable to express enjoyment.