answer:President Obama has proposed a 60 billion initiative to make community college free for the first two years, aiming to boost college access and workforce skills. The federal government would cover 75% of the cost, with states funding the remainder. Students would need to maintain good grades and be on track to complete their degrees. While the plan faces funding challenges, it builds on successful state programs like Tennessee’s Tennessee Promise. Maine's Sen. Angus King, who supports the idea, emphasized the importance of setting a national agenda and noted that similar proposals have spurred local action even without federal funding. Higher education experts and industry leaders welcome the focus on community colleges, which can provide essential skills and training for the modern economy. Despite potential hurdles in Congress, the proposal could inspire states to implement their own initiatives, addressing the skills gap and improving college affordability.

answer:This result is a fundamental property of binary representation, a cornerstone of computer science. To prove it, let's start by considering an arbitrary positive integer n. We aim to show that n can be expressed as a sum of distinct powers of 2. Begin by finding the largest power of 2 that is less than or equal to n, say 2^k. Such a k exists, since 2^0 = 1 is less than any positive integer. Now, we can write n as n = 2^k + m, where m is a non-negative integer. Since 2^k is the largest power of 2 less than or equal to n, m must be strictly less than 2^k. Now, apply the same reasoning to m: find the largest power of 2 less than or equal to m, say 2^j, and write m = 2^j + l. Continuing this process, we obtain a sequence of non-negative integers l, each smaller than the previous one, until we eventually reach 0. The key insight here is that each power of 2 used in the decomposition is distinct, since each subsequent remainder is strictly smaller than the previous power of 2. This ensures that no power of 2 is repeated in the sum. Formally, this process yields a representation of n as a sum of distinct powers of 2: n = 2^k + 2^j + ... + 2^0, where the exponents k, j, ..., 0 form a strictly decreasing sequence. This decomposition is, in fact, the binary representation of n, which is unique. This proof not only demonstrates the result but also provides an algorithm for finding the binary representation of a given positive integer. The uniqueness of the binary representation follows from the deterministic nature of the process: at each step, the largest power of 2 less than or equal to the remaining value is uniquely determined.

answer:The connection between the result and the infinite geometric series lies in the fact that the binary representation of a positive integer can be viewed as a finite sum of terms from the infinite geometric series 1 + x + x^2 + ..., where x = 1/2. To see this, recall the formula for the sum of an infinite geometric series: 1 + x + x^2 + ... = 1/(1 - x), valid for |x| < 1. When x = 1/2, the series becomes 1 + 1/2 + 1/4 + ..., and its sum is 1/(1 - 1/2) = 2. Now, consider a positive integer n with binary representation n = 2^k + 2^j + ... + 2^0. Dividing both sides by 2^k, we obtain n/2^k = 1 + 1/2^(k-j) + ... + 1/2^k. This expression can be viewed as a finite sum of terms from the infinite geometric series 1 + x + x^2 + ..., where x = 1/2. In other words, the binary representation of n corresponds to selecting a subset of terms from the infinite geometric series, where each term is either included (if the corresponding power of 2 is present in the binary representation) or excluded (if it's not). The sum of these selected terms yields the original number n. This connection highlights the intimate relationship between binary arithmetic and the geometric series 1 + x + x^2 + ..., demonstrating how the latter provides a mathematical framework for understanding the former. Furthermore, this perspective can be generalized to other bases, such as ternary or hexadecimal, by considering different values of x in the geometric series.

answer:The concept of expressing positive integers as a sum of distinct powers of 2 is closely related to the combinatorial concept of combinations, particularly through the notion of binary strings. Consider a binary string of length k, consisting of 0s and 1s. Each position in the string corresponds to a power of 2, from 2^0 to 2^(k-1). A 1 in the string indicates that the corresponding power of 2 is included in the sum, while a 0 indicates that it's not. Now, observe that the number of binary strings of length k with exactly r 1s is given by the binomial coefficient C(k, r), which represents the number of ways to choose r elements from a set of k distinct elements. This is because each binary string corresponds to a unique selection of r powers of 2 from the set {2^0, 2^1, ..., 2^(k-1)}. In other words, the binary representation of a positive integer n can be viewed as a selection of distinct powers of 2, which in turn can be represented as a binary string with 1s in the positions corresponding to the selected powers. The number of such selections, or binary strings, is precisely the number of combinations of k items taken r at a time, where k is the number of powers of 2 considered and r is the number of 1s in the binary string. To illustrate this connection, consider the number 11, which has binary representation 1011. This corresponds to selecting the powers 2^3, 2^1, and 2^0 from the set {2^0, 2^1, 2^2, 2^3}. There are C(4, 3) = 4 ways to select 3 powers of 2 from this set, and each selection corresponds to a unique binary string of length 4 with exactly 3 1s. This relationship highlights the combinatorial nature of binary arithmetic and demonstrates how the concept of expressing positive integers as a sum of distinct powers of 2 is intimately connected to the concept of combinations.