answer:To prove ((n+1)(n+2)(n+3)ldots(n+n)=2^{n}cdot1cdot3cdot5ldots(2n-1)) for (ninmathbb{N}^{*}) using mathematical induction, when transitioning from (n=k) to (n=k+1), the algebraic expression that needs to be multiplied on the left side is (dfrac{(k+1+k)(k+1+k+1)}{k+1}=2(2k+1)). Therefore, the answer is: (boxed{4k+2}). When transitioning from (n=k) to (n=k+1), the algebraic expression that needs to be multiplied on the left side is (dfrac{(k+1+k)(k+1+k+1)}{k+1}), which simplifies to the answer. This problem tests the application of mathematical induction, reasoning, and computational skills, and is considered a medium-level question.

answer:Answer: (0 ; 1),(1 ; 0).Solution. Subtract the first equation from the second:x^{2}(x-1)+y^{2}left(y^{3}-1right)=0 .From the first equation of the system, it follows that x leqslant 1 and y leqslant 1. Therefore, x^{2}(x-1) leqslant 0 and y^{2}left(y^{3}-1right) leqslant 0. The sum of two non-negative numbers is zero if and only if both are zero. If the first term in the left-hand side of equation (*) is zero, then x=0 or x=1. In the first case, from the original system, it follows that y=1. And in the second case, obviously, y=0.Evaluation. 14 points for a correct solution. If the answer is guessed but not proven that there are no other solutions, 2 points.

answer:Analysis: This problem is a typical application problem involving two unknowns, which can be easily solved by setting up equations. The key is to accurately identify the relationship between quantities. We set one unknown as x, and express the other unknown in terms of x, then set up and solve the equation accordingly.Let's denote the number of chickens as x. Based on the problem statement, we can express the number of ducks in terms of x, and then use the total number of chickens and ducks, which is 239, to set up an equation for solving.Let the number of chickens be x. According to the problem, the number of ducks is 3x + 15. Since the total number of chickens and ducks is 239, we can set up the following equation:x + (3x + 15) = 239Solving this equation, we get:4x + 15 = 239Subtracting 15 from both sides gives:4x = 224Dividing both sides by 4 yields:x = 56Therefore, the number of chickens is boxed{56}. To find the number of ducks, we substitute x = 56 into the expression for the number of ducks:3 times 56 + 15 = 168 + 15 = 183Hence, the number of ducks is boxed{183}.

answer:Since the common difference d neq 0, and a_1 = 2d,a_k is the geometric mean of a_1 and a_{2k+7},We have: (a_1 + (k-1)d)^2 = a_1(a_1 + (2k+6)d).Substitute a_1 = 2d into the equation:(2d + (k-1)d)^2 = 2d(2d + (2k+6)d).Expanding and simplifying the equation, we get:d^2(k^2-10k+9)=0.Since d neq 0, we can divide both sides by d^2:k^2 - 10k + 9 = 0.Solving the quadratic equation, we find that k = 5 or k = -3. However, since k represents the position in the sequence, we discard the negative value.Hence, the correct answer is: boxed{C}.To solve this problem, we applied the formula for the terms of an arithmetic sequence and the property of geometric means. This is a basic question that requires careful reading and understanding of the arithmetic sequence properties.

answer:# SolutionNotice that 345 and 5 y^{2} are divisible by 5, so 3 x^{2} must also be divisible by 5. Therefore, quad x=5 t, t in Z. Similarly, y=3 n, n in Z. After simplification, the equation becomes 5 t^{2}+3 n^{2}=23. Therefore, t^{2} leq frac{23}{5}, n^{2} leq frac{23}{3} or |t| leq 2,|n| leq 2. By trying the corresponding values of t, n, we get that |t|=2,|n|=1 orleft{begin{array}{c}x_{1}=-10 y_{1}=-3end{array},left{begin{array}{c}x_{2}=-10 y_{2}=3end{array},left{begin{array}{l}x_{3}=10 y_{3}=-3end{array},left{begin{array}{c}x_{4}=10 y_{4}=3end{array}right.right.right.right.The maximum value of the expression x+y is 10+3=13.Answer. 13.

answer:AnalysisThis question mainly examines the transformation rules of the graph of the function y=Asin (omega x+varphi). It is a basic question.By shifting the graph of the function y=sin 2x to the right by dfrac{pi}{6} units, the result can be obtained.SolutionSolution: By shifting the graph of the function y=sin 2x to the right by dfrac{pi}{6} units,we can obtain the graph of y=sin 2(x- dfrac{pi}{6} ),which is the graph of y=sin left(2x- dfrac{pi}{3}right), thus, the correct choice is boxed{A}.