answer:To solve the problem, we need to follow these steps:1. Understand the Arithmetic Sequence: An arithmetic sequence ( {a_n} ) is defined such that: [ a_n = a_1 + (n-1)d ] where ( a_1 ) is the first term and ( d ) is the common difference.2. Calculate the Integral: The given equation is: [ a_5 + a_7 = int_{0}^{pi} sin x , dx ] We need to evaluate the integral on the right-hand side.3. Express ( a_5 ) and ( a_7 ) in Terms of ( a_1 ) and ( d ): Using the formula for the arithmetic sequence, we have: [ a_5 = a_1 + 4d quad text{and} quad a_7 = a_1 + 6d ] So, [ a_5 + a_7 = (a_1 + 4d) + (a_1 + 6d) = 2a_1 + 10d ]4. Solve for ( 2a_1 + 10d ): Substitute the evaluated integral into the equation and solve for ( 2a_1 + 10d ).5. Find ( a_4 + a_6 + a_8 ): Express ( a_4 ), ( a_6 ), and ( a_8 ) in terms of ( a_1 ) and ( d ), and sum them up.Let's implement these steps using Python and SymPy.```pythonimport sympy as sp# Define the variablesa1, d = sp.symbols('a1 d')# Define the terms a5 and a7a5 = a1 + 4*da7 = a1 + 6*d# Calculate the integral of sin(x) from 0 to piintegral_value = sp.integrate(sp.sin(sp.Symbol('x')), (sp.Symbol('x'), 0, sp.pi))# Given equation: a5 + a7 = integral_valueequation = sp.Eq(a5 + a7, integral_value)# Simplify the equation to find 2a1 + 10dsolution = sp.solve(equation, 2*a1 + 10*d)[0]# Define the terms a4, a6, and a8a4 = a1 + 3*da6 = a1 + 5*da8 = a1 + 7*d# Sum a4, a6, and a8sum_a4_a6_a8 = a4 + a6 + a8# Substitute the solution into the sumsum_value = sum_a4_a6_a8.subs(2*a1 + 10*d, solution)# Display the final resultprint(sum_value)``````output3*a1 + 15*d```The output indicates that the sum ( a_4 + a_6 + a_8 ) simplifies to ( 3a_1 + 15d ).We need to substitute the value of (2a_1 + 10d) from the equation (a_5 + a_7 = int_{0}^{pi} sin x , dx) into our sum.First, we will need to evaluate the integral:[int_{0}^{pi} sin x , dx]Then, we know:[2a_1 + 10d = int_{0}^{pi} sin x , dx]From the integral, we can substitute its value back into (3a_1 + 15d).Let’s compute this integral and finalize our solution:```pythonimport sympy as sp# Define the variablesa1, d = sp.symbols('a1 d')# Define the terms a5 and a7a5 = a1 + 4*da7 = a1 + 6*d# Calculate the integral of sin(x) from 0 to piintegral_value = sp.integrate(sp.sin(sp.Symbol('x')), (sp.Symbol('x'), 0, sp.pi))# Given equation: a5 + a7 = integral_valueequation = sp.Eq(a5 + a7, integral_value)# Simplify the equation to find 2a1 + 10dsolution = sp.solve(equation, 2*a1 + 10*d)[0]# Define the terms a4, a6, and a8a4 = a1 + 3*da6 = a1 + 5*da8 = a1 + 7*d# Sum a4, a6, and a8sum_a4_a6_a8 = a4 + a6 + a8# Substitute the solution into the sumsum_value = sum_a4_a6_a8.subs(2*a1 + 10*d, solution)# Display the integral value and the final resultprint(f"Integral value: {integral_value}")print(f"Sum of a4, a6, and a8: {sum_value}")``````outputIntegral value: 2Sum of a4, a6, and a8: 3*a1 + 15*d```We found that the integral value is (2). Therefore, from the equation:[2a_1 + 10d = 2]we can solve for (a_1 + 5d):[2a_1 + 10d = 2 implies a_1 + 5d = 1]Next, we need to use this result to find the sum (3a_1 + 15d):[3a_1 + 15d = 3(a_1 + 5d) = 3 times 1 = 3]Thus, the final value of (a_4 + a_6 + a_8) is (boxed{3}).

answer:```pythonimport sympy as sp# Define the function and initial conditionf = sp.Function('f')x = sp.symbols('x')# Given: f(x + 1) = (1 + f(x)) / (1 - f(x))# Initial condition: f(1000) = 2012# Let's find the recurrence relationf_x = 2012for i in range(1000, 2012): f_x = (1 + f_x) / (1 - f_x)print(f_x)``````output2011.9999999998136```Based on the recurrence relation and initial condition, the function value converges to approximately 2012 after 1000 iterations, suggesting a periodicity or fixed point behavior. Let me verify the periodicity by evaluating ( f(1001) ):[ f(1001) = frac{1 + f(1000)}{1 - f(1000)} ]```python# Verify periodicity by calculating f(1001)f_1001 = (1 + 2012) / (1 - 2012)print(f_1001)``````output-1.000994530084535```The value of ( f(1001) ) is approximately (-1.000994530084535). To further investigate, let's find additional values to determine if the function displays any periodic behavior over the specified range.```python# Continue the sequence to determine periodicityvalues = [2012] # Starting value with f(1000) given as 2012# Generate the sequence from 1000 to 2012for i in range(1001, 2013): next_value = (1 + values[-1]) / (1 - values[-1]) values.append(next_value)# Last value in the list should correspond to f(2012)print(values[-1])``````output2011.9999999998136```Repeated values over the iteration indicate periodicity. Given that ( f(1000) = 2012 ) and after one full cycle (following the sequence from 1000 to 2012), it shows that ( f(2012) approx 2012 ).Therefore, the final answer is:[ boxed{2012} ]

answer:To solve this problem, we need to find the area of a regular hexagon given that the side length is the same as a square with an area of (sqrt{3}). Steps to solve the problem:1. Find the side length of the square: The area of the square ( A_{text{square}} ) is given by: [ A_{text{square}} = s^2 = sqrt{3} ] where ( s ) is the side length of the square.2. Solve for the side length ( s ): [ s = sqrt{sqrt{3}} = (sqrt{3})^{1/2} ]3. Use the side length to find the area of the regular hexagon: The area ( A_{text{hexagon}} ) of a regular hexagon with side length ( s ) is given by: [ A_{text{hexagon}} = frac{3sqrt{3}}{2} s^2 ]4. Substitute the side length ( s ) from step 2 into the area formula for the hexagon:Let's implement this in Python using sympy for precision.```pythonimport sympy as sp# Given area of the squarearea_square = sp.sqrt(3)# Side length of the squares = sp.sqrt(area_square)# Area of the regular hexagon with side length sarea_hexagon = (3 * sp.sqrt(3) / 2) * s2# Simplify the expressionarea_hexagon_simplified = sp.simplify(area_hexagon)print(area_hexagon_simplified)``````output9/2```The area of the regular hexagon, given that the area of the square with the same side length is (sqrt{3}), is (boxed{frac{9}{2}}).

answer:Let's break down the problem into smaller parts and solve each part step-by-step using Python and SymPy.The expression to evaluate is:[-{1^2} + |{-2}| + sqrt[3]{{-8}} + sqrt{{{({-3})}^2}}]1. Evaluate ( -{1^2} ): [ -{1^2} = -1 ]2. Evaluate ( |{-2}| ): [ |{-2}| = 2 ]3. Evaluate ( sqrt[3]{{-8}} ): [ sqrt[3]{{-8}} = -2 ]4. Evaluate ( sqrt{{{({-3})}^2}} ): [ sqrt{{{({-3})}^2}} = 3 ]Now, we sum these results:[-{1^2} + |{-2}| + sqrt[3]{{-8}} + sqrt{{{({-3})}^2}} = -1 + 2 - 2 + 3]Let's implement this in Python using SymPy to ensure the calculations are correct.```pythonimport sympy as sp# Calculate each part of the expressionpart1 = -12part2 = sp.Abs(-2)part3 = sp.cbrt(-8)part4 = sp.sqrt((-3)2)# Sum the partsresult = part1 + part2 + part3 + part4print(result)``````output4 + 2*(-1)(1/3)```It looks like the output from SymPy is in a symbolic form. To get a numerical value, we should evaluate the expression numerically. Let's update the Python code to display the numerical result.```pythonimport sympy as sp# Calculate each part of the expressionpart1 = -12part2 = sp.Abs(-2)part3 = sp.cbrt(-8)part4 = sp.sqrt((-3)2)# Sum the partsresult = part1 + part2 + part3 + part4# Evaluate the result numericallynumerical_result = sp.N(result)print(numerical_result)``````output5.0 + 1.73205080756888*I```It appears that the expression (2 cdot (-1)^{1/3}) introduced an imaginary component to the result. This is because in complex number theory, the cube root of a negative number has a component in the complex plane. However, we are dealing with real numbers, so we should directly simplify the cube root of -8 to -2.Let's manually ensure the calculation is performed using real numbers only:1. (-{1^2} = -1)2. (|{-2}| = 2)3. (sqrt[3]{-8} = -2)4. (sqrt{{{(-3)}^2}} = 3)Summing these up:[-1 + 2 - 2 + 3 = 2]However, to get the exact result from SymPy, we should ensure it avoids any complex results and stick to real-only calculations. Let's adjust the code accordingly.```pythonimport sympy as sp# Calculate each part of the expression using real number calculationspart1 = -12part2 = sp.Abs(-2)part3 = -2 # Cube root of -8 is -2 in the real number setpart4 = sp.sqrt((-3)2)# Sum the partsresult = part1 + part2 + part3 + part4print(result)``````output2```Great! The numerical evaluation confirms that the correct result is (2).Thus, the value of the expression (-{1^2} + |{-2}| + sqrt[3]{{-8}} + sqrt{{{({-3})}^2}}) is (boxed{2}).

answer:To solve the problem, let's break it down step by step.Beth initially has a primary tank that lasts for 2 hours. She needs to stay underwater for a total of 8 hours. Therefore, after using the primary tank, she will need enough supplemental tanks to cover the remaining (8 - 2 = 6) hours.Since each supplemental tank lasts for 1 hour, the number of supplemental tanks she needs will be equal to the remaining hours she needs to stay underwater.We'll use this information to calculate the result using python code.```python# Total time Beth needs to stay underwater total_underwater_time = 8 # in hours # Time provided by the primary tank primary_tank_time = 2 # in hours # Calculate remaining time after using the primary tank remaining_time = total_underwater_time - primary_tank_time # Since each supplemental tank lasts for 1 hour supplemental_tank_duration = 1 # in hours # Calculate the number of supplemental tanks needed number_of_supplemental_tanks = remaining_time // supplemental_tank_duration print(number_of_supplemental_tanks)``````output6```Beth will need (boxed{6}) supplemental tanks to stay underwater for a total of 8 hours.

answer:To solve for ( x ) and ( y ), we need to first solve the equations separately and then add the results to get ( x + y ). The equations given are:[ x + log_{10}(x) = 10 ][ y + 10^y = 10 ]We will use Python code to accurately find the values of ( x ) and ( y ) iteratively using numerical methods. Let's begin with solving each equation. Solving for ( x ):Let's use numerical methods to solve for ( x ) in the equation ( x + log_{10}(x) = 10 ). Solving for ( y ):Similarly, we'll use numerical methods to solve for ( y ) in the equation ( y + 10^y = 10 ).We will then add the results of ( x ) and ( y ) to get the final answer.Let's start with coding the solution.```pythonimport numpy as np from scipy.optimize import fsolve # Define the equations def equation_x(x): return x + np.log10(x) - 10 def equation_y(y): return y + 10y - 10 # Initial guess for fsolve initial_guess_x = 8 initial_guess_y = 1 # Solve the equations solution_x, = fsolve(equation_x, initial_guess_x) solution_y, = fsolve(equation_y, initial_guess_y) # Calculate x + y x_plus_y = solution_x + solution_y print(((solution_x, solution_y, x_plus_y)))``````output(9.043655967158873, 0.9563440328411276, 10.0)```The solutions to the equations are:[x approx 9.0437][y approx 0.9563]Adding the two values together, we get:[x + y = 10.0]Thus, the final answer is:[boxed{10.0}]