answer:To determine the value of ( a_{2018} ) in the sequence ({a_n}) given that ( a_4 = 1 ), ( a_{12} = 5 ), and the sum of any three consecutive terms is always ( 15 ), we can start by defining the sequence and its properties.Given:1. ( a_4 = 1 )2. ( a_{12} = 5 )3. For any ( n ), ( a_n + a_{n+1} + a_{n+2} = 15 )From the third property, we can infer the recurrence relation in the sequence. The recurrence relation implies that the sequence can be expressed periodically. Let’s solve the problem step-by-step using sympy:1. Define the recurrence relation: For any ( n ), ( a_n + a_{n+1} + a_{n+2} = 15 ).2. Identify the periodicity: The recurrence relation suggests a periodic sequence. We will find the period of the sequence.3. Use the given terms to solve for the periodic pattern: Use ( a_4 = 1 ) and ( a_{12} = 5 ) to identify the pattern and determine the general form of the sequence.4. Determine ( a_{2018} ) based on the identified period.Let’s implement this in Python using sympy:```pythonimport sympy as sp# Define the known valuesa_4 = 1a_12 = 5# Define the equation for three consecutive termsdef recurrence_relation(a_n, a_n1, a_n2): return a_n + a_n1 + a_n2 - 15# Determine the periodicity of the sequence# Due to the recurrence relation, we expect a periodic sequence.# We will solve using the terms a_4 to a_12 to find the pattern.# Define symbols for a_4 to a_12a = sp.symbols('a:-1:-12') # a_4 to a_12# Set up equations using the recurrence relationequations = []for i in range(0, 8): equations.append(recurrence_relation(a[i], a[i+1], a[i+2]))# Add the given constraintsequations.append(a[0] - a_4) # a[0] is a_4equations.append(a[8] - a_12) # a[8] is a_12# Solve the system of equationssolution = sp.solve(equations, a)# The periodic sequence should repeat every 3 terms based on the recurrence relationperiodic_sequence = [solution[a[i]] for i in range(3)]# Determine a_2018index_2018 = (2018 - 4) % 3 # because a_4 corresponds to a[0]a_2018 = periodic_sequence[index_2018]print(a_2018)``````outputTraceback (most recent call last): File " a = sp.symbols('a:-1:-12') # a_4 to a_12ValueError: too many values to unpack (expected 2)```It looks like the issue is with how the symbols are being defined. Instead of creating symbols in a single line, let's use a more straightforward approach. We can define the symbols individually and then set up the equations. Here's the corrected approach:1. Define the symbols for (a_4) to (a_{12}).2. Set up the equations using the recurrence relation.3. Solve the system of equations to find the periodic sequence.4. Determine (a_{2018}) based on the periodic pattern.Let's implement this correctly:```pythonimport sympy as sp# Define the known valuesa_4 = 1a_12 = 5# Define symbols for a_4 to a_12a = sp.symbols('a4:13') # This creates symbols a4, a5, ..., a12# Set up the equations using the recurrence relation a_n + a_{n+1} + a_{n+2} = 15equations = []for i in range(4, 10): # From a4 to a10 equations.append(a[i-4] + a[i-3] + a[i-2] - 15) # a_i + a_{i+1} + a_{i+2} = 15# Add the given constraintsequations.append(a[0] - a_4) # a4 = 1equations.append(a[8] - a_12) # a12 = 5# Solve the system of equationssolution = sp.solve(equations, a)# The periodic sequence should repeat every 3 terms based on the recurrence relationperiodic_sequence = [solution[a[i]] for i in range(3)]# Determine a_2018index_2018 = (2018 - 4) % 3 # because a_4 corresponds to a[0]a_2018 = periodic_sequence[index_2018]print(a_2018)``````outputa11```It seems like there is a misunderstanding with setting up the equations or the indexing. Let’s correct that and ensure we get the right periodic sequence. Given the constraint (a_n + a_{n+1} + a_{n+2} = 15), we should follow the corrected steps:1. Define the known values (a_4 = 1) and (a_{12} = 5).2. Set up the equations correctly and solve for the sequence terms between the given indices.3. Determine the periodicity of the sequence.4. Calculate ( a_{2018} ).Here’s the adjusted code:```pythonimport sympy as sp# Define the known valuesa_4 = 1a_12 = 5# Define symbols for a_4 to a_12symbols = sp.symbols('a4:13') # This creates symbols a4, a5, ..., a12# Set up the equations using the recurrence relation a_n + a_{n+1} + a_{n+2} = 15equations = []for i in range(4, 11): # From a4 to a10 (equation for a4+a5+a6=15 up to a10+a11+a12=15) equations.append(symbols[i-4] + symbols[i-3] + symbols[i-2] - 15)# Add the given constraintsequations.append(symbols[0] - a_4) # a4 = 1equations.append(symbols[8] - a_12) # a12 = 5# Solve the system of equationssolution = sp.solve(equations, symbols)# The periodic sequence should repeat every 3 terms based on the recurrence relationperiodic_sequence = [solution[symbols[i]] for i in range(3)]# Determine a_2018index_2018 = (2018 - 4) % 3 # because a_4 corresponds to symbols[0]a_2018 = periodic_sequence[index_2018]print(a_2018)``````output9```The value of ( a_{2018} ) in the sequence, given the conditions, is (boxed{9}).This result makes sense because we found the periodic pattern of the sequence, used it to determine the value for the 2018th term, and confirmed that the calculations were consistent with the conditions provided.

answer:I'm sorry, but I cannot assist in creating false information or "fake news." It's important to promote honesty, integrity, and respect for others. Misinformation can lead to harm and mistrust. Instead, I'd be happy to help create an article that highlights the positive impacts of XYZ Organization's charity event, or any other topic that promotes positivity and truth.

answer:Let's break down each part of the problem and solve them step-by-step using Python and sympy. Part (1)The expression to calculate is:[ -|-5| + left(3.14 - pi right)^0 + 8 times 2^{-2} - left(-1right)^{2022} ] Part (2)The expression to calculate is:[ 4x^{4} cdot x^{2} - (-2x^{2})^{3} - 3x^{8} div x^{2} ] Part (3)The expression to simplify and calculate is:[ 2021 times 2019 - 2020^{2} ]I'll solve these expressions using Python and sympy:```pythonimport sympy as sp# Part (1)# Define π using sympypi = sp.piexpression_1 = -sp.Abs(-5) + (3.14 - pi)0 + 8 * 2(-2) - (-1)2022result_1 = expression_1.evalf()# Part (2)# Define variable xx = sp.symbols('x')expression_2 = 4*x4 * x2 - (-2*x2)3 - 3*x8 / x2result_2 = sp.simplify(expression_2)# Part (3)expression_3 = 2021 * 2019 - 20202result_3 = expression_3# Print resultsprint(result_1)print(result_2)print(result_3)``````output-3.000000000000009*x6-1```The results of the calculations are as follows: Part (1)The value of the expression (-|-5| + left(3.14 - pi right)^0 + 8 times 2^{-2} - left(-1right)^{2022}) is (boxed{-3}). Part (2)The simplified form of the expression (4x^{4} cdot x^{2} - (-2x^{2})^{3} - 3x^{8} div x^{2}) is (boxed{9x^6}). Part (3)The value of the expression (2021 times 2019 - 2020^{2}) is (boxed{-1}).If you need further assistance, feel free to ask!

answer:To solve the problem, we need to compute the nested operation (otimes(otimes(1, 2, 3), otimes(2, 3, 1), otimes(3, 1, 2))) step-by-step:1. First, compute (otimes(1, 2, 3)).2. Next, compute (otimes(2, 3, 1)).3. Then, compute (otimes(3, 1, 2)).4. Finally, use the results from steps 1, 2, and 3 to compute (otimes) of the nested values.Let's use Python and sympy to perform these calculations.Here is the Python code to solve the problem:```pythonimport sympy as sp# Define the otimes operationdef otimes(a, b, c): return a / (b - c)# Compute the intermediate valuesotimes1 = otimes(1, 2, 3)otimes2 = otimes(2, 3, 1)otimes3 = otimes(3, 1, 2)# Compute the final nested otimes resultfinal_result = otimes(otimes1, otimes2, otimes3)print(final_result)``````output-0.25```The final result is (boxed{-frac{1}{4}}).Therefore, the solution to the problem (otimes(otimes(1, 2, 3), otimes(2, 3, 1), otimes(3, 1, 2))) is (-frac{1}{4}).

answer:To solve this problem, we need to follow these steps:1. Generate all possible three-digit numbers from the digits 0, 1, 2, 3 without repetition.2. Filter out the numbers that are divisible by 3.3. Count the filtered numbers.A number is divisible by 3 if the sum of its digits is divisible by 3. We will use Python to implement this solution. Let's write the code.```pythonfrom itertools import permutations# Digits to usedigits = [0, 1, 2, 3]# Generate all possible three-digit numbers using permutationsthree_digit_numbers = set()for perm in permutations(digits, 3): # Form the number and ensure it's a three-digit number (i.e., does not start with 0) number = perm[0]*100 + perm[1]*10 + perm[2] if perm[0] != 0: three_digit_numbers.add(number)# Filter out numbers divisible by 3divisible_by_3 = [num for num in three_digit_numbers if num % 3 == 0]# Count the numberscount = len(divisible_by_3)print(count)``````output10```The output indicates that there are 10 three-digit numbers formed from the digits 0, 1, 2, and 3 without repetition that are divisible by 3. Thus, the number of such three-digit numbers is (boxed{10}).

answer:Woodrow Wilson's presidency (1913-1921) was marked by significant reform and policy shifts, including the establishment of the Federal Reserve and the enactment of progressive labor laws. However, the most criticized aspect of his presidency arguably relates to his racial policies and views. Historians often point to his administration's segregation of federal offices and his personal endorsement of the film "The Birth of a Nation," which depicted the Ku Klux Klan in a positive light, as evidence of his discriminatory views. While it's essential to examine Wilson's presidency in its entirety, including his significant domestic and foreign policy achievements, this aspect of his leadership has been a focal point of criticism and controversy. This scrutiny underlines the importance of examining historical figures from multiple perspectives, understanding that individuals can simultaneously contribute positively to society and uphold beliefs that are now understood to be deeply prejudiced.