answer:To solve the problem, let's break it down into manageable steps and use Python with SymPy to verify our results. Step 1: Prove that ({a_n}) is a quasi-arithmetic sequenceGiven:1. (a_1 = a)2. (a_n + a_{n+1} = 2n)We need to show that (a_{n+2} - a_n = d) for some constant (d).First, let's express (a_{n+1}) in terms of (a_n):[ a_{n+1} = 2n - a_n ]Next, express (a_{n+2}) in terms of (a_{n+1}):[ a_{n+2} = 2(n+1) - a_{n+1} = 2n + 2 - (2n - a_n) = a_n + 2 ]Thus, we have:[ a_{n+2} - a_n = (a_n + 2) - a_n = 2 ]So, the common difference (d) is (2), and ({a_n}) is indeed a quasi-arithmetic sequence with (d = 2). Step 2: Find the general formula for ({a_n}) and the sum (S_{20})The recurrence relation is:[ a_{n+2} = a_n + 2 ]We already know:[ a_1 = a ][ a_2 = 2 - a ]Using the recurrence relation, we can write:[ a_3 = a_1 + 2 = a + 2 ][ a_4 = a_2 + 2 = (2 - a) + 2 = 4 - a ][ a_5 = a_3 + 2 = (a + 2) + 2 = a + 4 ][ a_6 = a_4 + 2 = (4 - a) + 2 = 6 - a ]We observe the pattern:[ a_{2k-1} = a + 2(k-1) ][ a_{2k} = 2k - a ]Thus, the general formula is:[ a_n = begin{cases} a + 2left(frac{n-1}{2}right) = a + n - 1 & text{if } n text{ is odd} 2left(frac{n}{2}right) - a = n - a & text{if } n text{ is even}end{cases} ]Next, let's find the sum (S_{20}):[ S_{20} = sum_{n=1}^{20} a_n ]Let's implement this in Python using SymPy to ensure the results are accurate.```pythonimport sympy as sp# Define n and the first term an = sp.symbols('n')a = sp.symbols('a')# General formula for odd and even na_odd = a + n - 1a_even = n - a# Sum of the first 20 termsS_20 = 0for i in range(1, 21): if i % 2 == 0: S_20 += a_even.subs(n, i) else: S_20 += a_odd.subs(n, i)S_20 = sp.simplify(S_20)print(S_20)``````output200```The sum of the first 20 terms of the sequence ({a_n}) is (boxed{200}).To summarize the results:1. We proved that ({a_n}) is a quasi-arithmetic sequence with a common difference (d = 2).2. The general formula for ({a_n}) is: [ a_n = begin{cases} a + n - 1 & text{if } n text{ is odd} n - a & text{if } n text{ is even} end{cases} ]3. The sum of the first 20 terms (S_{20}) is (boxed{200}).

answer:Let's solve the problem step-by-step using Python and SymPy:1. We know that ( (x, y) = (2, -1) ) is a solution to the linear equation ( ax + by = 2 ).2. Substitute ( x = 2 ) and ( y = -1 ) into the equation ( ax + by = 2 ) to find the relationship between ( a ) and ( b ).3. Solve for ( a ) in terms of ( b ).4. Use this relationship to calculate the value of ( 2a - b - 4 ).Let's write the Python code to perform these steps:```pythonimport sympy as sp# Define the variablesa, b = sp.symbols('a b')# Given x and y valuesx_val = 2y_val = -1# Define the linear equation ax + by = 2equation = a * x_val + b * y_val - 2# Solve the equation for a in terms of ba_solution = sp.solve(equation, a)[0]# Define the expression 2a - b - 4expression = 2 * a_solution - b - 4# Simplify the expressionsimplified_expression = sp.simplify(expression)# Print the resultprint(simplified_expression)``````output-2```The value of (2a - b - 4) is (boxed{-2}).Therefore, the correct answer is:B: (-2)

answer:To solve this problem, we need to find the value of (a) such that the line passing through points (A(1+a, 2a)) and (B(1-a, 3)) has an inclination of (90^circ). The angle of inclination (90^circ) indicates that the line is vertical, implying that the x-coordinates of points (A) and (B) must be equal. Let's break down this problem step-by-step:1. Understand the Condition: The angle of inclination (90^circ) means that the line is vertical. Hence, the x-coordinates of points (A) and (B) must be the same.2. Set Up the Equation: Given points: [ A(1+a, 2a) quad text{and} quad B(1-a, 3) ] For the line to be vertical: [ 1 + a = 1 - a ]3. Solve for (a): Solve the above equation for (a) to find the required value.Let's implement this in Python using SymPy to solve for (a).```pythonimport sympy as sp# Define the variablea = sp.symbols('a')# Define the equation 1 + a = 1 - aequation = sp.Eq(1 + a, 1 - a)# Solve the equation for asolution = sp.solve(equation, a)# Print the solutionprint(solution)``````output[0]```The solution to the equation (1 + a = 1 - a) is (a = 0).Therefore, the value of (a) for which the line passing through points (A(1+a, 2a)) and (B(1-a, 3)) has an angle of inclination of (90^circ) is (boxed{0}).

answer:To solve this problem, we need to determine the length of the segment (AB) given the coordinates of points (A) and (B) and the fact that (AB) is parallel to the (y)-axis. Steps:1. Understand the Condition: - Since (AB) is parallel to the (y)-axis, the (x)-coordinates of points (A) and (B) must be equal. - This gives us the equation (3 = a).2. Substitute the Value of (a): - Substitute (a = 3) into the coordinates of points (A) and (B).3. Calculate the Length of Segment (AB): - The length of (AB) is the absolute difference between the (y)-coordinates of (A) and (B).Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the variablea = sp.symbols('a')# Coordinates of points A and BA_x = 3A_y = a + 3B_x = aB_y = a - 4# Condition for AB to be parallel to the y-axiscondition = sp.Eq(A_x, B_x)# Solve for 'a'a_solution = sp.solve(condition, a)[0]# Substitute 'a' in the y-coordinates of A and BA_y_value = A_y.subs(a, a_solution)B_y_value = B_y.subs(a, a_solution)# Calculate the length of segment ABlength_AB = sp.Abs(A_y_value - B_y_value)# Print the length of segment ABprint(length_AB)``````output7```The length of segment (AB) is (boxed{7}). Here is the breakdown of the solution:1. Since (AB) is parallel to the (y)-axis, the (x)-coordinates of points (A) and (B) must be equal. This gives us (3 = a).2. Substituting (a = 3) into the coordinates: - The coordinates of point (A) are ((3, a+3) = (3, 3+3) = (3, 6)). - The coordinates of point (B) are ((a, a-4) = (3, 3-4) = (3, -1)).3. The length of segment (AB) is the absolute difference between the (y)-coordinates of points (A) and (B): [ |6 - (-1)| = |6 + 1| = 7 ]Therefore, the length of segment (AB) is (boxed{7}).

answer:To solve this problem, we need to determine the number of ways to select and order 4 programs out of 8 options (A, B, C, D, E, F, G, H) while satisfying the given constraints. This can be broken down into clear steps:1. Select 4 Programs: - We need to select 4 programs out of the 8, ensuring that at least one of A or B is selected.2. Ensure A or B (or both) are Selected: - We need to calculate the total number of ways to choose 4 programs out of 8. - From this, subtract the cases where neither A nor B is selected.3. Ensure A and B are Not Adjacent: - If both A and B are selected, we need to ensure they are not adjacent in the performance order.Let's implement this in Python using sympy to handle the combinatorial calculations.1. Initial Setup: - Calculate the total number of ways to choose 4 programs from 8. - Subtract the number of ways to select 4 programs from the 6 options excluding A and B.2. Calculate Non-Adjacent Arrangements: - If both A and B are chosen, calculate the number of ways to arrange them such that they are not adjacent.Here's the Python code to solve this:```pythonimport sympy as spfrom itertools import combinations, permutations# Define the total number of programsprograms = ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H']# Calculate combinations of 8 programs taken 4 at a timetotal_combinations = list(combinations(programs, 4))# Filter combinations to ensure at least one of A or B is selectedvalid_combinations = [comb for comb in total_combinations if 'A' in comb or 'B' in comb]# Function to check if A and B are adjacentdef are_adjacent(order): for i in range(len(order) - 1): if (order[i] == 'A' and order[i + 1] == 'B') or (order[i] == 'B' and order[i + 1] == 'A'): return True return False# Calculate valid permutations for each valid combinationvalid_permutations = []for comb in valid_combinations: comb_permutations = list(permutations(comb)) for perm in comb_permutations: if ('A' in perm and 'B' in perm and not are_adjacent(perm)) or ('A' not in perm or 'B' not in perm): valid_permutations.append(perm)# Output the number of valid permutationsprint(len(valid_permutations))``````output1140```The number of different performance orders that satisfy the given constraints is (boxed{1140}).

answer:To decompose the vector ( x ) in terms of the vectors ( p, q, r ), we need to express ( x ) as a linear combination of ( p, q, r ). That is, we need to find scalars ( a, b, c ) such that:[ x = a p + b q + c r ]Given:[ x = {3, 1, 8} ][ p = {0, 1, 3} ][ q = {1, 2, -1} ][ r = {2, 0, -1} ]We can set up a system of linear equations based on the components of these vectors:[ begin{cases}a cdot 0 + b cdot 1 + c cdot 2 = 3 a cdot 1 + b cdot 2 + c cdot 0 = 1 a cdot 3 + b cdot (-1) + c cdot (-1) = 8end{cases} ]This can be written in matrix form as:[ begin{pmatrix}0 & 1 & 2 1 & 2 & 0 3 & -1 & -1end{pmatrix}begin{pmatrix}a b cend{pmatrix}=begin{pmatrix}3 1 8end{pmatrix} ]We will solve this system using Python and sympy.Let's write the Python code to solve this system:```pythonimport sympy as sp# Define the symbolsa, b, c = sp.symbols('a b c')# Define the vectorsx = sp.Matrix([3, 1, 8])p = sp.Matrix([0, 1, 3])q = sp.Matrix([1, 2, -1])r = sp.Matrix([2, 0, -1])# Setup the equationseq1 = sp.Eq(a * p[0] + b * q[0] + c * r[0], x[0])eq2 = sp.Eq(a * p[1] + b * q[1] + c * r[1], x[1])eq3 = sp.Eq(a * p[2] + b * q[2] + c * r[2], x[2])# Solve the system of equationssolution = sp.solve([eq1, eq2, eq3], (a, b, c))print(solution)``````output{a: 3, b: -1, c: 2}```The decomposition of the vector ( x ) in terms of the vectors ( p, q, r ) is given by:[ x = 3p - 1q + 2r ]Thus, the scalars ( a, b, c ) are ( boxed{3, -1, 2} ).