answer:To find (u_{16}) in terms of (a), we can start by examining the recursive sequence defined as:[u_{n + 1} = -frac{1}{u_n + 1}]with the initial condition (u_1 = a > 0). We will write a Python code to iteratively compute the values of this sequence up to (u_{16}).Let's define the sequence computation and iterate it 15 times starting from (u_1 = a).```python# Define the initial conditiona = 1 # You can change this value to any positive number au_n = a# Continuously update the sequence values up to u_16for _ in range(1, 16): u_n = -1 / (u_n + 1)print(u_n)``````output1.0```The value obtained for (u_{16}) is 1.0 when (a = 1). To generalize this, let's consider different values of (a). We will analyze the iterative behavior by adjusting (a) and see if there is a pattern or limit as (n) approaches 16. Let's compute and verify the attraction point for several initial conditions of (a).```python# Define a function to compute u_n def sequence(u, steps): for _ in range(1, steps + 1): u = -1 / (u + 1) return u # Test the function with different values of a and get u_16 results = {} for a in [0.5, 1, 2, 3, -0.5, -2]: results[a] = sequence(a, 15) print(results)``````output{0.5: 0.5000000000000001, 1: 1.0, 2: 2.000000000000001, 3: 3.000000000000001, -0.5: -0.5, -2: -2.0}```The results of (u_{16}) based on different initial values of (a) show that for any (a), (u_{16} = a). This suggests that the sequence defined by[u_{n + 1} = -frac{1}{u_n+1}]exhibits a cyclical pattern that returns to the original value after a certain number of iterations. It can be verified further by deeper analysis, but the observed values confirm:[boxed{u_{16} = a}]This means that the sequence returns to its initial value (a) after 15 iterations, thus (u_{16}) is simply (a).

answer:To solve the equation (log_{2} frac{1}{4} + log_{2} 32) using Python and SymPy, we can follow these steps:1. Import the required library.2. Define the logarithmic expressions.3. Simplify the expressions.4. Sum the simplified expressions.5. Print the final result.Let's write the Python code to perform these steps.```pythonimport sympy as sp# Define the base of the logarithmbase = 2# Define the expressionslog1 = sp.log(1/4, base)log2 = sp.log(32, base)# Sum the logarithmic expressionsresult = log1 + log2# Simplify the resultsimplified_result = sp.simplify(result)# Print the final resultprint(simplified_result)``````output(-1.38629436111989 + log(32))/log(2)```It looks like the output wasn't fully simplified. Let's break down the logarithmic expressions manually to make sure they simplify correctly, and then sum them up.We know:[ log_2 frac{1}{4} = log_2 2^{-2} = -2 ][ log_2 32 = log_2 2^5 = 5 ]Therefore:[ log_2 frac{1}{4} + log_2 32 = -2 + 5 = 3 ]Let's verify this with Python and SymPy by ensuring we simplify the logarithmic expressions correctly:```pythonimport sympy as sp# Define the base of the logarithmbase = 2# Define the expressionslog1 = sp.log(1/4, base)log2 = sp.log(32, base)# Simplify the logarithmic expressionssimplified_log1 = sp.simplify(log1)simplified_log2 = sp.simplify(log2)# Sum the simplified expressionsresult = simplified_log1 + simplified_log2# Print the final resultprint(result)``````output5 - 1.38629436111989/log(2)```It seems that the output hasn't simplified completely due to the division by (log(2)). Let's use SymPy to explicitly convert these logarithms to their numerical values and sum them up.SymPy can evaluate the logarithms more effectively using the `evalf()` function which evaluates the floating-point approximation. However, knowing the properties of logarithms and base-2 logarithm transformations should give the correct results as well.Let's directly simplify the expression more accurately:```pythonimport sympy as sp# Define the base of the logarithmbase = 2# Using SymPy's log and simplify functionslog1 = sp.log(1/4, base)log2 = sp.log(32, base)# Convert to numerical valuessimplified_log1 = log1.evalf()simplified_log2 = log2.evalf()# Sum the numerical valuesresult = simplified_log1 + simplified_log2# Print the final resultprint(result)``````output3.00000000000000```The output confirms our manual calculation that (log_{2} frac{1}{4} + log_{2} 32 = 3).Thus, the final answer is:[boxed{3}]

answer:To solve this problem, let's break it down into manageable steps:1. Understand the Sequence Properties: - The sequence is an arithmetic sequence. - The median of the sequence is 1010. - The last term of the sequence is 2015.2. Properties of an Arithmetic Sequence: Let's denote the first term by (a) and the common difference by (d). The general form of an arithmetic sequence is (a, a+d, a+2d, ldots, a+(n-1)d), where (n) is the number of terms.3. Determine the Median: Since the sequence has a median of 1010, there are an odd number of terms. The median of an arithmetic sequence with an odd number of terms is the middle term, which is the (left(frac{n+1}{2}right))th term.4. Formulate the Equations: - The last term is (a + (n-1)d = 2015). - The median term is (a + left(frac{n-1}{2}right)d = 1010).5. Solve the Equations: We need to find the values of (a) and (d) that satisfy these conditions. Now, let's use Python and sympy to find the first term (a).```pythonimport sympy as sp# Define the symbolsa, d, n = sp.symbols('a d n')# Equations based on the problem statementmedian_eq = a + ((n - 1) / 2) * d - 1010last_term_eq = a + (n - 1) * d - 2015# Solve the equations simultaneouslysolution = sp.solve((median_eq, last_term_eq), (a, d))# Extracting the value of the first term 'a'first_term = solution[a]print(first_term)``````output5```The first term of the arithmetic sequence is (boxed{5}).

answer:Let's break down the problem step by step, and then we'll check our reasoning using Python code.1. Total Arrests Calculation: - There are 21 cities. - Each city has 10 arrests per day. - Protests last for 30 days. Total number of arrests = (21 text{ cities} times 10 text{ arrests/day} times 30 text{ days})2. Time Spent in Jail Before Trial: - Each person spends 4 days in jail before their trial.3. Time Spent Serving Sentence: - Each person serves half of a 2-week sentence. - Half of 2 weeks = 1 week.4. Total Jail Time Calculation: - For each person, the total time in jail is their pre-trial time plus their sentence time. - Total time in jail for each person = (4 text{ days} + 1 text{ week}) - Convert 4 days to weeks: 4 days = (frac{4}{7}) weeks (since 1 week = 7 days).5. Combined Jail Time: - Multiply the total jail time per person by the total number of arrests.Let's do these calculations in Python to ensure accuracy.```python# Constants cities = 21 arrests_per_day = 10 protest_days = 30 pre_trial_jail_days = 4 sentence_half_weeks = 1 # Half of a 2-week sentence is 1 week # Calculations total_arrests = cities * arrests_per_day * protest_days pre_trial_jail_weeks = pre_trial_jail_days / 7 # Convert days to weeks total_jail_weeks_per_person = pre_trial_jail_weeks + sentence_half_weeks total_combined_jail_weeks = total_arrests * total_jail_weeks_per_person print(total_combined_jail_weeks)``````output9900.0```The total combined jail time, when converted into weeks, for all the arrests that occurred during the protests is (boxed{9900}) weeks.

answer:To solve this problem, we will use Python and the sympy library. Let's break down the problem step by step:1. Define Variables: - Let ( S ) be the traffic flow on the Second Ring Road. - Let ( T ) be the traffic flow on the Third Ring Road. - Let ( F ) be the traffic flow on the Fourth Ring Road.2. Formulate Equations Based on Student Reports: - From Student A: ( S = 10000 ) - From Student B: ( F = T + 2000 ) - From Student C: ( 3T - F = 2S )3. Substitute and Solve: - Substitute ( S = 10000 ) into the equations. - Use the equations ( F = T + 2000 ) and ( 3T - F = 2S ) to solve for ( T ) and ( F ).Let's implement this in Python using sympy to find the traffic flow on the Third Ring Road and the Fourth Ring Road.```pythonimport sympy as sp# Define the variablesS, T, F = sp.symbols('S T F')# Define the equations based on the student reportsequation1 = sp.Eq(S, 10000)equation2 = sp.Eq(F, T + 2000)equation3 = sp.Eq(3*T - F, 2*S)# Substitute the value of S into the equationsequation3 = equation3.subs(S, 10000)# Solve the system of equationssolution = sp.solve((equation1, equation2, equation3), (S, T, F))# Output the solutionprint(solution)``````output{F: 13000, S: 10000, T: 11000}```Based on the solution from the sympy code, we have the traffic flow values as follows:- The traffic flow on the Second Ring Road, ( S ), is ( 10000 ) vehicles per hour.- The traffic flow on the Third Ring Road, ( T ), is ( 11000 ) vehicles per hour.- The traffic flow on the Fourth Ring Road, ( F ), is ( 13000 ) vehicles per hour.Therefore, the traffic flow on the Third Ring Road and the Fourth Ring Road during peak hours are:[boxed{11000}]for the Third Ring Road and[boxed{13000}]for the Fourth Ring Road.

answer:Let's solve the problem step-by-step and use Python to verify our steps.Given the equation:[log_{25}(x-4) = frac{1}{2}]we can convert it to its exponential form to solve for ( x ):[25^{frac{1}{2}} = x - 4]First, let's solve the equation ( 25^{frac{1}{2}} ):[25^{frac{1}{2}} = sqrt{25} = 5]So,[5 = x - 4]Adding 4 to both sides, we get:[x = 9]Now we need to find:[frac{1}{log_{x}3}]Since ( x = 9 ), this can be written as:[frac{1}{log_{9}3}]Using the change of base formula for logarithms:[log_{9}3 = frac{log_{10}3}{log_{10}9}]We need to find the value of (log_{10}3) and (log_{10}9), then calculate the result.Let's use Python to accurately do these computations.```pythonimport math# Calculate the logarithmslog_3_base_10 = math.log10(3)log_9_base_10 = math.log10(9)# Calculate log_9(3) using the change of base formulalog_9_3 = log_3_base_10 / log_9_base_10# Calculate the final resultresult = 1 / log_9_3print(result)``````output2.0```The value of (frac{1}{log_{9}3}) is (boxed{2}).