answer:To solve the given expression, we can break it down into parts and simplify each part step-by-step using Python and SymPy. The expression is:[ {9}^{frac{1}{2}log_{3}10}+(frac{1}{81})^{frac{1}{4}}-0.027^{-frac{1}{3}}+log_{6}2+frac{1}{1+log_{3}2} ]Let's solve each term one by one:1. ({9}^{frac{1}{2}log_{3}10}): We know that (9 = 3^2), so this can be rewritten as: [ 9^{frac{1}{2}log_{3}10} = (3^2)^{frac{1}{2}log_{3}10} = 3^{2 cdot frac{1}{2} log_{3}10} = 3^{log_{3}10} = 10 ]2. ((frac{1}{81})^{frac{1}{4}}): We know that (81 = 3^4), so this can be rewritten as: [ (frac{1}{3^4})^{frac{1}{4}} = (3^{-4})^{frac{1}{4}} = 3^{-1} = frac{1}{3} ]3. (0.027^{-frac{1}{3}}): We know that (0.027 = 27 times 10^{-3} = (3^3 times 10^{-3}) = 3^{-3}), so: [ 0.027^{-frac{1}{3}} = (3^{-3})^{-frac{1}{3}} = 3 ]4. (log_{6}2): This term can be left as is since it represents the logarithm function with base 6.5. (frac{1}{1+log_{3}2}): This term can also be left as is and calculated directly.Now, let's write Python code using SymPy to solve this expression and get the final value.```pythonimport sympy as sp# Define the expression componentsexpr1 = 9(0.5 * sp.log(10, 3))expr2 = (1/81)(1/4)expr3 = 0.027(-1/3)expr4 = sp.log(2, 6)expr5 = 1 / (1 + sp.log(2, 3))# Sum the components to get the final resultfinal_expression = expr1 + expr2 - expr3 + expr4 + expr5# Calculate the final resultfinal_value = sp.N(final_expression)print(final_value)``````output8.00000000000000```The value of the given expression ({9}^{frac{1}{2}log_{3}10}+(frac{1}{81})^{frac{1}{4}}-0.027^{-frac{1}{3}}+log_{6}2+frac{1}{1+log_{3}2}) is (boxed{8}).

answer:To solve this problem, we need to find out which single bag was delivered to customer C, given that the total weight of oranges delivered to customer A is twice that of customer B. Let's break down the solution into smaller parts:1. Define the problem variables: - We need to find the weight of the single bag delivered to customer C. - The remaining 8 bags will be distributed between customers A and B such that the total weight of the bags delivered to A is twice the total weight of the bags delivered to B.2. Set up equations: - Let ( C ) be the weight of the bag delivered to customer C. - Let ( T_A ) be the total weight of the bags delivered to customer A. - Let ( T_B ) be the total weight of the bags delivered to customer B. We have: [ T_A = 2 times T_B ] The sum of all weights is: [ sum = 22 + 25 + 28 + 31 + 34 + 36 + 38 + 40 + 45 = 299 text{ kg} ]3. Constraints: - ( T_A + T_B + C = 299 ) - ( T_A = 2 times T_B ) - We need to find ( C ) such that the remaining weights can be split into two parts with the required ratio.Let's implement this in Python using sympy:```pythonimport sympy as spfrom itertools import combinations# List of weightsweights = [22, 25, 28, 31, 34, 36, 38, 40, 45]# Total sum of all weightstotal_sum = sum(weights)# Function to find the weight of the bag delivered to customer Cdef find_weight_for_customer_C(weights, total_sum): # Iterate through each bag to check if it can be the bag for customer C for weight_C in weights: remaining_weights = [weight for weight in weights if weight != weight_C] for k in range(1, len(remaining_weights)): for A_weights in combinations(remaining_weights, k): T_A = sum(A_weights) T_B = total_sum - T_A - weight_C if T_A == 2 * T_B: return weight_C return Noneweight_C = find_weight_for_customer_C(weights, total_sum)print(weight_C)``````output38```The weight of the bag of oranges delivered to customer C is (boxed{38}) kilograms.

answer:To solve this problem, we need to determine the common ratio of the geometric sequence and then find the value of the sixth term.Let's break down the steps:1. Understand the problem: We need to insert six numbers between 6 and 768 to form a geometric sequence with a total of eight terms. 2. Form the geometric sequence: A geometric sequence has the form (a, ar, ar^2, ar^3, ldots, ar^{n-1}) where (a) is the first term and (r) is the common ratio.3. Identify the known terms and common ratio: We know the first term (a = 6) and the eighth term (ar^7 = 768).4. Solve for the common ratio (r): Using (ar^7 = 768), we get (6r^7 = 768). Solve for (r).5. Find the sixth term: The sixth term of the sequence is (ar^5).Let's implement this in Python using sympy to solve for (r) and find the sixth term.```pythonimport sympy as sp# Define the variablesa = 6 # first termeighth_term = 768 # eighth term as ar^7# Define symbol for common ratior = sp.symbols('r')# Create the equation ar^7 = 768equation = a * r7 - eighth_term# Solve the equation for rcommon_ratio = sp.solve(equation, r)[0]# Calculate the sixth term ar^5sixth_term = a * common_ratio5# Simplify and print the sixth termsixth_term_simplified = sp.simplify(sixth_term)print(sixth_term_simplified)``````output192```The sixth term of the geometric sequence that starts with 6 and ends with 768, with six inserted numbers, is (boxed{192}).

answer:To solve the problem, let's break it down into two main tasks:1. Find the general term formula for the geometric sequence ({a_n}).2. Determine the minimum value of (m in mathbb{Z}) such that (S_n < m) always holds true. Step 1: General Term Formula for ({a_n})Since ({a_n}) is a geometric sequence with common ratio (q < 1), the general term (a_n) can be expressed as:[ a_n = a_1 cdot q^{n-1} ]Given:- (a_2 = 2)- (S_3 = 7)Using the geometric sequence properties:[ a_2 = a_1 cdot q^{1} = a_1 cdot q = 2 implies a_1 = frac{2}{q} ]The sum of the first (n) terms of a geometric sequence is given by:[ S_n = a_1 frac{1 - q^n}{1 - q} ]For (n = 3):[ S_3 = a_1 frac{1 - q^3}{1 - q} = 7 ]Substitute (a_1 = frac{2}{q}):[ frac{2}{q} frac{1 - q^3}{1 - q} = 7 ]Solve for (q). Step 2: Minimum Value of (m) such that (S_n < m) always holds trueOnce we find the value of (q), we'll compute the sum (S_n) for an arbitrary (n) and determine the minimum integer (m) such that (S_n < m) for all (n).Let's implement this step-by-step in Python using sympy:```pythonimport sympy as sp# Define the variableq = sp.symbols('q')# Given conditionsa2 = 2S3 = 7# First, find a1 in terms of qa1 = a2 / q# Sum of the first 3 termsS3_expr = a1 * (1 - q3) / (1 - q)# Solve for q given S3 = 7q_value = sp.solve(S3_expr - 7, q)q_value = [sol.evalf() for sol in q_value if sol < 1 and sol > 0]# Extract the valid q valueq_valid = q_value[0]# Calculate a1 using the valid q valuea1_value = a2 / q_valid# General term formula a_na_n = a1_value * q_valid(sp.symbols('n') - 1)# Define the sum of the first n terms S_nn = sp.symbols('n')S_n = a1_value * (1 - q_validn) / (1 - q_valid)# Simplify S_nS_n_simplified = sp.simplify(S_n)# Find the minimum m such that S_n < m always holds true# Since q < 1, we consider the limit of S_n as n approaches infinityS_infinity = sp.limit(S_n_simplified, n, sp.oo)# Find the smallest integer greater than S_infinitymin_m = sp.ceiling(S_infinity)min_m = min_m.evalf()print(f"q = {q_valid}")print(f"a1 = {a1_value}")print(f"a_n = {a_n}")print(f"S_infinity = {S_infinity}")print(f"Minimum value of m = {min_m}")``````outputq = 0.500000000000000a1 = 4.00000000000000a_n = 4.0*0.5(n - 1)S_infinity = 8Minimum value of m = 8.00000000000000```From the output of the Python code, we have the following values:1. The common ratio ( q ) is ( 0.5 ).2. The first term ( a_1 ) is ( 4 ).3. The general term formula for the sequence ({a_n}) is: [ a_n = 4 cdot 0.5^{n-1} ]4. The sum of the terms as ( n ) approaches infinity ( S_{infty} ) is ( 8 ).Since ( S_n < m ) always holds true, the minimum value of ( m ) must be greater than or equal to ( S_{infty} ). Therefore, the minimum value of ( m ) that satisfies ( S_n < m ) for all ( n ) is ( 8 ).Thus, the minimum value of ( m ) is:[boxed{8}]

answer:To solve this problem, we need to use the properties of arithmetic sequences. Let's break down the problem step-by-step:1. Define the arithmetic sequence: An arithmetic sequence is defined by its first term (a_1) and the common difference (d). The (n)-th term of an arithmetic sequence can be written as: [ a_n = a_1 + (n-1)d ]2. Given condition: We are given that (a_7 + a_8 + a_9 = 3).3. Express the terms in the condition: Using the formula for the (n)-th term: [ a_7 = a_1 + 6d, quad a_8 = a_1 + 7d, quad a_9 = a_1 + 8d ] Therefore: [ a_7 + a_8 + a_9 = (a_1 + 6d) + (a_1 + 7d) + (a_1 + 8d) = 3a_1 + 21d ] Given that this sum is equal to 3, we have: [ 3a_1 + 21d = 3 ] 4. Solve for (a_1) and (d): Simplify the equation: [ a_1 + 7d = 1 ]5. Sum of the first 15 terms: The sum of the first (n) terms of an arithmetic sequence is given by: [ S_n = frac{n}{2} left(2a_1 + (n-1)dright) ] For (n = 15): [ S_{15} = frac{15}{2} left(2a_1 + 14dright) ]Now, let's solve for (a_1) and (d) and then compute (S_{15}) using Python and sympy.```pythonimport sympy as sp# Define variablesa1, d = sp.symbols('a1 d')# Given equationeq = sp.Eq(a1 + 7*d, 1)# Solve for a1 in terms of dsol = sp.solve(eq, a1)[0]# Define the sum of the first 15 termsn = 15S_n = (n / 2) * (2*a1 + (n-1)*d)# Substitute the solution into the sum formulaS_15 = S_n.subs(a1, sol)# Simplify the expressionS_15_simplified = sp.simplify(S_15)print(S_15_simplified)``````output15.0000000000000```The sum of the first 15 terms of the arithmetic sequence, given that (a_7 + a_8 + a_9 = 3), is (boxed{15}).The sum of the first 15 terms of the arithmetic sequence is (boxed{15}).

answer:To solve this problem, we need to analyze the function ( f(x) ), which is defined as the minimum of the three functions ( x + 2 ), ( 4x + 1 ), and (-2x + 4 ). Here's the step-by-step approach:1. Identify the regions where each function is the minimum: We need to find the points where the functions ( x + 2 ), ( 4x + 1 ), and (-2x + 4 ) intersect, as these points will help us determine the intervals where each function is the minimum.2. Solve for the intersection points: Set ( x + 2 = 4x + 1 ), ( x + 2 = -2x + 4 ), and ( 4x + 1 = -2x + 4 ) to find the critical points.3. Analyze ( f(x) ) in different intervals: Determine the minimum function in each interval formed by the critical points and find the maximum value of ( f(x) ) over these intervals.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the variable and functionsx = sp.symbols('x')f1 = x + 2f2 = 4*x + 1f3 = -2*x + 4# Solve for intersection pointssol1 = sp.solve(f1 - f2, x)[0]sol2 = sp.solve(f1 - f3, x)[0]sol3 = sp.solve(f2 - f3, x)[0]# Determine the intervalsintervals = sorted([sol1, sol2, sol3])# Function to find the minimum function in each intervaldef find_minimum_in_interval(x_val): return min(f1.subs(x, x_val), f2.subs(x, x_val), f3.subs(x, x_val))# Evaluate f(x) at critical points and midpoints of intervalsvalues = [find_minimum_in_interval(intervals[0]-1), find_minimum_in_interval((intervals[0]+intervals[1])/2), find_minimum_in_interval(intervals[1]), find_minimum_in_interval((intervals[1]+intervals[2])/2), find_minimum_in_interval(intervals[2]), find_minimum_in_interval(intervals[2]+1)]# Find the maximum value among these evaluationsmax_value = max(values)print(max_value)``````output8/3```The maximum value of the function ( f(x) ), which is defined as the minimum of the three functions ( x + 2 ), ( 4x + 1 ), and (-2x + 4 ), is (boxed{frac{8}{3}}).