answer:To solve the problem, we'll follow these steps:1. Identify the geometric sequence properties: - General term of the geometric sequence is given by (a_n = a cdot r^{n-1}), where (a) is the first term and (r) is the common ratio. - Given (a_3 = 4) and (a_4 = -8), we can use these to determine (a) and (r).2. Formulate equations based on the given terms: - (a cdot r^2 = 4) - (a cdot r^3 = -8)3. Solve for (a) and (r): - Divide the second equation by the first to eliminate (a).4. Sum of the first (n) terms of a geometric sequence: - The sum (S_n) of the first (n) terms is given by (S_n = a frac{r^n - 1}{r - 1}) for (r neq 1).5. Use the values of (a) and (r) to find (S_5).Let's implement this step-by-step in Python using SymPy:```pythonimport sympy as sp# Define the variablesa, r = sp.symbols('a r')# Given valuesa3 = 4a4 = -8# Equations based on the given termseq1 = a * r2 - a3eq2 = a * r3 - a4# Solve the system of equationssolution = sp.solve((eq1, eq2), (a, r))a_value, r_value = solution[a], solution[r]# Calculate S_5n = 5S_n = a * (rn - 1) / (r - 1)S_5 = S_n.subs({a: a_value, r: r_value})print(S_5)``````outputTraceback (most recent call last): File " a_value, r_value = solution[a], solution[r]TypeError: list indices must be integers or slices, not Symbol```It appears there was an issue with how the solutions were unpacked. The `sp.solve` function returns a list of dictionaries, and we should access the first dictionary to retrieve the values of `a` and `r`.Let's make the necessary adjustments:1. Ensure we correctly access the values of `a` and `r` from the solution.2. Substitute these values into the formula for (S_5).Here is the corrected code:```pythonimport sympy as sp# Define the variablesa, r = sp.symbols('a r')# Given valuesa3 = 4a4 = -8# Equations based on the given termseq1 = a * r2 - a3eq2 = a * r3 - a4# Solve the system of equationssolution = sp.solve((eq1, eq2), (a, r))solution_dict = solution[0] # Access the first solution dictionarya_value, r_value = solution_dict[a], solution_dict[r]# Calculate S_5n = 5S_n = a * (rn - 1) / (r - 1)S_5 = S_n.subs({a: a_value, r: r_value})print(S_5)``````outputTraceback (most recent call last): File " a_value, r_value = solution_dict[a], solution_dict[r]TypeError: tuple indices must be integers or slices, not Symbol```The error suggests that the solution is returned as a list of tuples rather than a dictionary. We need to adjust the code to handle this correctly.Let's fix this by unpacking the solution correctly:```pythonimport sympy as sp# Define the variablesa, r = sp.symbols('a r')# Given valuesa3 = 4a4 = -8# Equations based on the given termseq1 = a * r2 - a3eq2 = a * r3 - a4# Solve the system of equationssolution = sp.solve((eq1, eq2), (a, r))# Unpack the solution(a_value, r_value) = solution[0]# Calculate S_5n = 5S_n = a * (rn - 1) / (r - 1)S_5 = S_n.subs({a: a_value, r: r_value})print(S_5)``````output11```The value of (S_5) is (boxed{11}).Here is a summary of the steps taken:1. We identified the properties of the geometric sequence and formulated the equations based on the given terms.2. We solved the system of equations to find the values of (a) and (r).3. We used these values to calculate the sum of the first 5 terms of the sequence, (S_5).Thus, the sum of the first 5 terms of the geometric sequence is (boxed{11}).

answer:A geometric progression (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio."Given:- The second term ( T_2 = 5 )- The third term ( T_3 = 1 )Let's denote:- The first term by ( a )- The common ratio by ( r )From the properties of a geometric progression, we have:[ T_2 = ar ][ T_3 = ar^2 ]Using the given values:[ ar = 5 ][ ar^2 = 1 ]We can solve these equations to find ( a ) and ( r ). First, we can solve for ( r ) by dividing the second equation by the first equation:[ frac{ar^2}{ar} = frac{1}{5} ][ r = frac{1}{5} ]Next, we substitute ( r ) back into the first equation to solve for ( a ):[ a left(frac{1}{5}right) = 5 ][ a = 5 times 5 ][ a = 25 ]Thus, the first term ( a ) is 25.Let's verify this with Python to ensure the result is accurate.```python# Given valuesT2 = 5 # second termT3 = 1 # third term# common ratior = T3 / T2# first terma = T2 / rprint((a, r))``````output(25.0, 0.2)```The computations confirm our reasoning:- The common ratio ( r = frac{1}{5} = 0.2 )- The first term ( a = 25 )Thus, the first term of the geometric progression is (boxed{25}).

answer:To solve the problem of tiling a ( 1993 times 2000 ) rectangle using ( 2 times 2 ) squares, ( P )-rectangles, and ( S )-rectangles in such a way that the sum of the number of ( 2 times 2 ) squares and ( S )-rectangles used (denoted by ( s )) is maximized, we can start by examining the tiling capacities of each piece.1. A ( 2 times 2 ) square covers 4 cells.2. A ( P )-rectangle is a ( 2 times 3 ) or ( 3 times 2 ) rectangle with one corner cell missing, leaving 5 cells covered.3. An ( S )-rectangle is a ( 2 times 3 ) or ( 3 times 2 ) rectangle with two opposite corner cells missing, leaving 4 cells covered.Firstly, let's determine the total number of cells in the ( 1993 times 2000 ) rectangle:[ text{Total cells} = 1993 times 2000 ]Next, our goal is to maximize the usage of ( 2 times 2 ) squares and ( S )-rectangles. Note that each ( 2 times 2 ) square and each ( S )-rectangle covers exactly 4 cells. Therefore, each square or ( S )-rectangle contributes equally to the total cell count, and we need to maximize the number of these pieces.We will also consider ( P )-rectangles separately, which we use as few as possible since they contribute to the sum ( s ).To cover the cells optimally:1. Let ( a ) be the number of ( 2 times 2 ) squares used.2. Let ( b ) be the number of ( S )-rectangles used.3. Let ( c ) be the number of ( P )-rectangles used.[4a + 4b + 5c = 1993 times 2000]Our objective is to maximize ( s = a + b ).We will write a Python program to solve this optimizing problem.```python# Define the dimensions of the large rectanglewidth = 1993height = 2000# Total number of cells in the large rectangletotal_cells = width * height# The number of cells each type of rectangle coverscells_per_square = 4cells_per_s_rectangle = 4cells_per_p_rectangle = 5# Function to calculate the maximum sdef calculate_max_s(total_cells, cells_per_square, cells_per_s_rectangle, cells_per_p_rectangle): max_s = 0 # Iterate over possible values of b (number of S-rectangles) for b in range(total_cells // cells_per_s_rectangle + 1): # Remaining cells after using b S-rectangles remaining_cells = total_cells - b * cells_per_s_rectangle # Try to use as many 2x2 squares as possible with the remaining cells a = remaining_cells // cells_per_square # Check if we can exactly cover remaining cells with 5-cells P-rectangles if (remaining_cells - a * cells_per_square) % cells_per_p_rectangle == 0: # Find the number of P-rectangles c = (remaining_cells - a * cells_per_square) // cells_per_p_rectangle # Sum of squares and S-rectangles s = a + b # Update max_s if we get a larger sum if s > max_s: max_s = s max_a = a max_b = b max_c = c return max_s, max_a, max_b, max_c# Calculate the maximal smaximal_s, max_a, max_b, max_c = calculate_max_s(total_cells, cells_per_square, cells_per_s_rectangle, cells_per_p_rectangle)print((maximal_s, max_a, max_b, max_c))``````output(996500, 996500, 0, 0)```The maximal value of ( s ), the sum of the number of ( 2 times 2 ) squares and ( S )-rectangles used, is given by:[ boxed{996500} ]This is achieved by using:- ( 996500 ) ( 2 times 2 ) squares,- ( 0 ) ( S )-rectangles,- and ( 0 ) ( P )-rectangles.

answer:To solve this problem, we need to find the number of positive four-digit integers that are divisible by 8.The range of four-digit integers is between 1000 and 9999.- The smallest four-digit integer is 1000.- The largest four-digit integer is 9999.An integer is divisible by 8 if the last three digits form a number that is divisible by 8.Our goal is to count how many numbers between 1000 and 9999, inclusive, are divisible by 8.We can write a Python script to iterate through this range and count the numbers that satisfy the condition. Let's proceed with the code implementation.```pythoncount = 0# iterate over each number in the four-digit rangefor number in range(1000, 10000): if number % 8 == 0: count += 1print(count)``````output1125```The number of positive four-digit integers that are divisible by (8) is (boxed{1125}).

answer:Let's define and solve the problem step-by-step using Python and SymPy.First, let's break down the problem:1. Define the Arithmetic Sequence: The general form of an arithmetic sequence can be expressed as: [ a_n = a_1 + (n - 1)d ] where (a_1) is the first term and (d) is the common difference.2. Given Equation: We are given: [ a_1 + a_{13} = 10 ] We can express (a_{13}) in terms of (a_1) and (d) as: [ a_{13} = a_1 + 12d ] So the equation becomes: [ a_1 + (a_1 + 12d) = 10 ] Simplifying this, we get: [ 2a_1 + 12d = 10 ]3. Finding the Required Sum: We need to find the value of (a_3 + a_5 + a_7 + a_9 + a_{11}). We can express each term in terms of (a_1) and (d): [ a_3 = a_1 + 2d, quad a_5 = a_1 + 4d, quad a_7 = a_1 + 6d, quad a_9 = a_1 + 8d, quad a_{11} = a_1 + 10d ] Summing these: [ a_3 + a_5 + a_7 + a_9 + a_{11} = (a_1 + 2d) + (a_1 + 4d) + (a_1 + 6d) + (a_1 + 8d) + (a_1 + 10d) ] [ = 5a_1 + (2d + 4d + 6d + 8d + 10d) ] [ = 5a_1 + 30d ]4. Solve for (a_1) and (d): Using the given equation (2a_1 + 12d = 10), we can solve for (a_1) and (d).Let's implement this in Python using SymPy to find the required sum.```pythonimport sympy as sp# Define variablesa_1, d = sp.symbols('a_1 d')# Given equationequation = sp.Eq(2*a_1 + 12*d, 10)# Solve for a_1 and dsolution = sp.solve(equation, (a_1, d))a_1_value = solution[a_1]d_value = solution[d]# Calculate the required sumsum_terms = 5*a_1 + 30*drequired_sum = sum_terms.subs({a_1: a_1_value, d: d_value})# Print the resultprint(required_sum)``````outputTraceback (most recent call last): File " a_1_value = solution[a_1]TypeError: list indices must be integers or slices, not Symbol```It looks like there was an error in accessing the solution. The `sp.solve` function returns a list of dictionaries when solving multiple variables. We need to handle this correctly.Let's update the code to correctly access the values of (a_1) and (d) from the solution. Here is the corrected implementation:```pythonimport sympy as sp# Define variablesa_1, d = sp.symbols('a_1 d')# Given equationequation = sp.Eq(2*a_1 + 12*d, 10)# Solve for a_1 and dsolution = sp.solve(equation, (a_1, d))# Access the solutionsa_1_value = solution[0][a_1]d_value = solution[0][d]# Calculate the required sumsum_terms = 5*a_1 + 30*drequired_sum = sum_terms.subs({a_1: a_1_value, d: d_value})# Print the resultprint(required_sum)``````outputTraceback (most recent call last): File " a_1_value = solution[0][a_1]TypeError: tuple indices must be integers or slices, not Symbol```I apologize for the oversight. When solving simultaneous equations, `sympy.solve` returns a list of tuples. We need to correctly unpack the tuple to obtain (a_1) and (d).Let's fix the code accordingly:```pythonimport sympy as sp# Define variablesa_1, d = sp.symbols('a_1 d')# Given equationequation = sp.Eq(2*a_1 + 12*d, 10)# Solve for a_1 and dsolution = sp.solve(equation, (a_1, d))# Access the solutionsa_1_value, d_value = solution[0]# Calculate the required sumsum_terms = 5*a_1 + 30*drequired_sum = sum_terms.subs({a_1: a_1_value, d: d_value})# Print the resultprint(required_sum)``````output25```The value of (a_3 + a_5 + a_7 + a_9 + a_{11}) for the given arithmetic sequence is (boxed{25}).

answer:To solve the problem, we need to simplify the given expression and then substitute ( x = 1 ) and ( y = -2 ) to find its value.Given expression:[ 3y^2 - x^2 + 2(2x^2 - 3xy) - 3(x^2 + y^2) ]Let's break this down into smaller steps:1. Simplify the expression.2. Substitute ( x = 1 ) and ( y = -2 ) into the simplified expression.3. Evaluate the simplified expression to find the value.Let's implement this in Python using sympy.```pythonimport sympy as sp# Define the variablesx, y = sp.symbols('x y')# Define the expressionexpression = 3*y2 - x2 + 2*(2*x2 - 3*x*y) - 3*(x2 + y2)# Simplify the expressionsimplified_expression = sp.simplify(expression)# Substitute x = 1 and y = -2value = simplified_expression.subs({x: 1, y: -2})print(value)``````output12```The value of the expression (3y^2 - x^2 + 2(2x^2 - 3xy) - 3(x^2 + y^2)) when (x = 1) and (y = -2) is (boxed{12}).