answer:1. Consider the equation in the problem: [ 2 - 2^{-|x-2|} = 2 + a ] We need to find the range of the real number ( a ) such that this equation has a real root ( x ). 2. Begin by examining the term ( 2^{-|x-2|} ): [ -|x-2| leq 0 Rightarrow 2^{-|x-2|} in (0, 1] ] Here, ( 2^{-|x-2|} ) takes values between 0 (exclusive) and 1 (inclusive). 3. Subtracting ( 2^{-|x-2|} ) from 2: [ 2 - 2^{-|x-2|} in [1, 2) ] This is because when ( 2^{-|x-2|} = 1 ), the expression ( 2 - 2^{-|x-2|} ) equals 1; and as ( 2^{-|x-2|} ) approaches 0, ( 2 - 2^{-|x-2|} ) approaches 2. 4. We are given the equation: [ 2 - 2^{-|x-2|} = 2 + a ] From the interval found above, we have: [ 1 leq 2 + a < 2 ] 5. Now solve for ( a ) in the inequality: [ 1 leq 2 + a < 2 ] Subtract 2 from all parts of the inequality: [ 1 - 2 leq 2 + a - 2 < 2 - 2 ] Simplifying each part gives: [ -1 leq a < 0 ] 6. Therefore, the range of the real number ( a ) that allows the equation to have a real root ( x ) is: [ -1 leq a < 0 ] Conclusion: (boxed{text{D}})

answer:Start by denoting the terms of the arithmetic progression as 9, 9+d, and 9+2d. After adding 5 to the second term and 25 to the third term, they become 9, 14+d, and 34+2d, respectively. Since these must form a geometric progression: [ (14+d)^{2} = 9(34 + 2d). ] Expanding and simplifying, we get: [ 196 + 28d + d^2 = 306 + 18d. ] [ d^2 + 10d - 110 = 0. ] This quadratic equation factors as: [ (d - 10)(d + 11) = 0. ] Thus, d = 10 or d = -11. For d = 10, the geometric sequence is 9, 24, 54, and for d = -11, the sequence is 9, 3, -3. The smallest possible value for the third term of the geometric progression is thus boxed{-3}.

answer:Given that A(2, n-1) is a point on the graph of the inverse proportion function y=frac{k}{x} where kneq 0, we can derive several conclusions step by step. 1. Since A(2, n-1) lies on the graph, we have the equation for point A as n-1 = frac{k}{2}. This implies that k=2(n-1). Since kneq 0, it follows that n-1 neq 0, which means n neq 1. Therefore, option A, which suggests that the value of n may be 1, is incorrect. 2. For point C to be on the graph of the inverse proportion function, its coordinates (n-1, n) must satisfy the equation n = frac{k}{n-1}. Substituting k=2(n-1) into this equation gives us (n-1)n = 2(n-1). Since we know n-1 neq 0, we can divide both sides of the equation by n-1, leading to n=2. This means that k=2times(2-1)=2, and the coordinates of point C become (1, 2). Therefore, point C can indeed be on the graph of the inverse proportion function, making option B incorrect. 3. When considering the behavior of the graph of the inverse proportion function y=frac{k}{x}, if k<0, which happens when n-1<0, the function shows that y increases as x increases on one branch of the graph. This confirms that option C is correct. 4. If we consider the case when n=0, the line BC would lie on the x-axis. This scenario shows that the line BC does not necessarily intersect the graph of the inverse proportion function y=frac{k}{x}, making option D incorrect. Therefore, after analyzing each option based on the given information and the properties of the inverse proportion function, we conclude that the correct option is: boxed{text{C}}.

answer:1. **Calculate the total sum of all integers from 1 to 49:** The sum of the first n natural numbers is given by the formula: [ S = frac{n(n+1)}{2} ] Here, n = 49. [ S = frac{49 times 50}{2} = 1225 ] 2. **Determine the number of rows, columns, and diagonals:** Since the matrix is 7 times 7, there are 7 rows, 7 columns, and 2 main diagonals. 3. **Calculate the common sum for each row, column, and diagonal:** Since the sum of all numbers in the matrix is 1225 and there are 7 rows, the sum of the numbers in each row must be: [ text{Sum per row} = frac{text{Total sum}}{text{Number of rows}} = frac{1225}{7} = 175 ] This sum also applies to each column and each diagonal because the matrix is arranged such that all rows, columns, and diagonals sum to the same value. 4. **Conclusion:** The value of the common sum for each row, column, and diagonal in the 7 times 7 square matrix is 175. The final answer is boxed{textbf{(C) } 175}