answer:Let's find the pattern of multiples of 3 that sum up to 72, with the largest number being 27. Since 27 is the largest number and it is a multiple of 3, we can start by listing the multiples of 3 leading up to 27: 3, 6, 9, 12, 15, 18, 21, 24, 27 Now, let's add these numbers together to see if their sum is 72: 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135 The sum of these numbers is 135, which is greater than 72. Therefore, not all of these multiples can be part of the pattern. We need to find a subset of these multiples that add up to 72. Let's start with the largest number, 27, and work our way down to see which combination of these multiples will give us the sum of 72. Starting with 27, let's subtract it from 72 to see how much we need to reach the sum: 72 - 27 = 45 Now we need to find a combination of the remaining multiples that add up to 45. Let's try adding the next largest multiple, 24: 45 - 24 = 21 We now have 21 left to reach the sum of 72. The next largest multiple of 3 that is less than or equal to 21 is 21 itself: 21 - 21 = 0 We have reached the sum of 72 with the combination of 27, 24, and 21. Therefore, the pattern of multiples of 3 that adds up to 72 is: 21, 24, 27 The sum of these numbers is indeed 72: 21 + 24 + 27 = boxed{72}

answer:Using the formula for binomial coefficients, binom{n}{k} = frac{n!}{k!(n-k)!}, we compute binom{12}{3}: [ binom{12}{3} = frac{12!}{3!(12-3)!} = frac{12 times 11 times 10}{3 times 2 times 1} = frac{1320}{6} = 220 ] Conclusion: The answer is boxed{220}.

answer:Transforming the general formula of the sequence, we have a_n = frac{n - sqrt{2015}}{n - sqrt{2016}} = 1 + frac{sqrt{2016} - sqrt{2015}}{n - sqrt{2016}}. This function is decreasing in both intervals (0, sqrt{2016}) and (sqrt{2016}, +infty). Since 44 < sqrt{2016} < 45, Therefore, among the first 50 terms of this sequence, the largest and smallest terms are a_{45} and a_{44}, respectively. Hence, the correct choice is boxed{text{D}}. By transforming the given general formula of the sequence and considering a_n as a function of n, we obtain the answer. This problem examines the functional characteristics of sequences and the idea of combining numerical and geometric methods. The key to solving this problem is to draw a graph based on the general formula of the sequence, making it a fundamental question.

answer:Let the page numbers be n and n + 1. We then have the equation: n(n+1) = 20412. Simplifying this, we get: n^2 + n - 20412 = 0. To solve for n, use the quadratic formula: n = frac{-b pm sqrt{b^2 - 4ac}}{2a}, where a = 1, b = 1, and c = -20412. Plugging in these values: n = frac{-1 pm sqrt{1^2 + 4 times 1 times 20412}}{2 times 1} = frac{-1 pm sqrt{81649}}{2} = frac{-1 pm 287}{2}. Solving for n, we get two values: n = frac{-1 + 287}{2} = 143 and n = frac{-1 - 287}{2} = -144. Since page numbers must be positive, we choose n = 143. Therefore, the next page is n + 1 = 144. Thus, the sum of these two page numbers is: n + (n + 1) = 143 + 144 = boxed{287}.