answer:Given that the square roots of a positive number are x and x-6, we aim to find this positive number. Step 1: Understand that the square roots of a positive number are given as x and x-6. Step 2: Set up the equation based on the given information. Since these are the square roots of the same positive number, squaring either should give us the positive number. Therefore, we can set up the equation as follows: [x^2 = (x-6)^2] Step 3: Expand the equation to find x. [x^2 = x^2 - 12x + 36] Step 4: Simplify the equation. Notice that x^2 on both sides cancels out, leaving us with: [0 = -12x + 36] Step 5: Solve for x. [12x = 36] [x = 3] Step 6: Substitute x = 3 back into the expressions for the square roots to verify or to find the positive number. [x = 3 Rightarrow (3)^2 = 9] [x-6 = 3-6 = -3 Rightarrow (-3)^2 = 9] Both square roots, when squared, give us the positive number 9. Therefore, the positive number is boxed{9}. 【Analysis】: By understanding the definition of square roots and setting up an equation based on the given square roots, we were able to find the value of x and subsequently the positive number itself. 【Comment】: This problem tests the understanding of square roots and the ability to manipulate algebraic expressions to find the value of an unknown that satisfies given conditions.

answer:Since in the arithmetic sequence {a_n}, a_5+a_8+a_{11}=3a_8 is a constant, it follows that a_8 is a constant. Furthermore, a_8 is the arithmetic mean of a_1 and a_{15}, therefore, S_{15}=15a_8 is a constant. Hence, the answer is S_{15}. Thus, the constant value among S_n is boxed{S_{15}}.

answer:To determine which of the given numbers is irrational, we evaluate each option based on the properties of rational and irrational numbers. - **Option A: dfrac{22}{7}** This is a fraction where both the numerator and the denominator are integers. According to the definition, a rational number can be expressed as a fraction dfrac{a}{b} where a and b are integers, and b neq 0. Therefore, dfrac{22}{7} is a rational number. - **Option B: 0.3** This is a decimal number that terminates. Terminating decimals can be expressed as fractions, making them rational numbers. Specifically, 0.3 can be written as dfrac{3}{10}, which is a fraction of two integers. Hence, 0.3 is a rational number. - **Option C: dfrac{pi }{2}** pi is known to be an irrational number because it cannot be expressed as a fraction of two integers. Its decimal representation is non-terminating and non-repeating. Since the multiplication or division of an irrational number by a non-zero rational number results in an irrational number, dfrac{pi }{2} is irrational. This makes option C the correct answer. - **Option D: 0.10101ldots 101** This is a repeating decimal, indicated by the ellipsis (...). Repeating decimals can be expressed as fractions, which means they are rational numbers. For example, 0.10101ldots can be expressed as a series sum or using a specific formula that results in a fraction. Given the above evaluations: - Option A is rational. - Option B is rational. - Option C is irrational. - Option D is rational. Therefore, the correct answer, which is the irrational number among the options, is boxed{text{Option C}}.

answer:1. **Original angles setup**: Start with the angles angle A_0 = 60.2^circ, angle B_0 = 59.8^circ, and angle C_0 = 60^circ. 2. **Recurrent relations**: - For each triangle triangle A_{n-1}B_{n-1}C_{n-1}, altitudes are drawn changing the character of triangle dynamics. Each alteration cycle brings an incremental angle change through: [ x_{n+1} = 180^circ - 2x_n, ; y_{n+1} = 180^circ - 2y_n, ; z_{n+1} = 180^circ - 2z_n. ] 3. **Analysis for right triangle condition**: - We need to find when x_n = 90^circ or any of the other angles becomes 90 degrees. - The equations become: [ begin{align*} x_n &= 180^circ - 2x_{n-1} x_1 &= 180^circ - 2 times 60.2^circ = 59.6^circ x_2 &= 180^circ - 2 times 59.6^circ = 60.8^circ x_3 &= 180^circ - 2 times 60.8^circ = 58.4^circ text{and so on...} end{align*} ] Iterate until an angle close to 90^circ appears. 4. Calculate for y_n and z_n similarly. 5. Conclusion: Check until which iteration x_n, y_n or z_n approx 90^circ. Suppose at position n = 12, one of the angles first equals (or surpasses) 90^circ. n = 12 The final answer is boxed{C) 12}