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question:The complex number with the imaginary part of sqrt{2}+7i as the real part and the real part of sqrt{7}i+5i^{2} as the imaginary part is ( ) A: 75i B: sqrt{2}+sqrt{7}i C: 5+i D: sqrt{2}+sqrt{7}i
answer:To solve for the complex number described, we start by understanding the structure of a complex number, which is generally denoted as z = a + bi, where a is the real part and b is the imaginary part, and both a and b are real numbers. Given that the imaginary part of -sqrt{2}+7i is to be the real part of our complex number, we identify the imaginary part as 7. Thus, we have: [a = 7] Next, we're told that the real part of sqrt{7}i + 5i^{2} is to be the imaginary part of our complex number. To find the real part of this expression, we first simplify it. Since i^{2} = -1, we have: [sqrt{7}i + 5(-1) = sqrt{7}i - 5] The real part of this expression is -5, so: [b = -5] Therefore, the complex number z can be written as: [z = a + bi = 7 - 5i] Hence, the correct answer is: [boxed{A: 7-5i}]
question:Let's play a number game: Step 1: Take a natural number, n_1 = 5, calculate n_1^2 + 1 to get a_1; Step 2: Calculate the sum of the digits of a_1 to get n_2, then calculate n_2^2 + 1 to get a_2; Step 3: Calculate the sum of the digits of a_2 to get n_3, then calculate n_3^2 + 1 to get a_3; And so on, then a_{2009} =
answer:Since n_1 = 5, we have a_1 = 26, n_2 = 8, a_2 = 65, n_3 = 11, a_3 = 122, n_4 = 5, ..., Therefore, a_{2009} = a_2 = 65. Hence, the answer is: boxed{65}.
question:A certain company has records stored with a record storage firm in 15inch by 12inch by 10inch boxes. The boxes occupy 1.08 million cubic inches of space. The company pays a certain amount per box per month for the record storage, and the total amount that the company pays each month for record storage is 300. How much does the company pay per box per month for the record storage?
answer:First, let's calculate the volume of one box: Volume of one box = length × width × height Volume of one box = 15 inches × 12 inches × 10 inches Volume of one box = 1800 cubic inches Now, let's find out how many boxes are stored: Total volume of all boxes = 1.08 million cubic inches Number of boxes = Total volume of all boxes / Volume of one box Number of boxes = 1,080,000 cubic inches / 1800 cubic inches Number of boxes = 600 Now that we know the number of boxes, we can calculate the cost per box per month: Total cost per month = 300 Cost per box per month = Total cost per month / Number of boxes Cost per box per month = 300 / 600 Cost per box per month = 0.50 Therefore, the company pays boxed{0.50} per box per month for the record storage.
question:f proposition P(n) satisfies: (1) P(n) is true for n=2; (2) If P(n) is true for n=k(kin mathbb{N}^*), then it is also true for n=k+2, then the correct conclusion is ( ) A: P(n) is true for all positive integers n B: P(n) is true for all positive even numbers n C: P(n) is true for all positive odd numbers n D: P(n) is true for all positive integers n greater than 1
answer:**Analysis** This question mainly examines the application of mathematical induction, with the key being the correct use of the induction hypothesis. By utilizing the established hypothesis for k=2, it is deduced that n being a positive even number always holds, leading to the conclusion. **Answer** Given that proposition P(n) is true for n=k(kin mathbb{N}^*), it then also holds true for n=k+2. If P(n) is true for n=2, then it is also true for n=4, 6, 8, …, 2m, which means P(n) is true for all positive even numbers n, therefore, the correct choice is boxed{text{B}}.