answer：Okay, so I have this equation: |x - 2| + |x + 1| = B, and I need to find how many integers x satisfy this equation. The answer will be C, and I need to figure out what C is. Hmm, let me think about how to approach this.First, I remember that absolute value equations can sometimes be tricky because they involve distances on the number line. Maybe I should consider different cases based on the value of x. The absolute value expressions |x - 2| and |x + 1| change their behavior depending on whether x is greater than or less than 2 and -1, respectively.So, let me identify the critical points where the expressions inside the absolute values change sign. For |x - 2|, the critical point is at x = 2, and for |x + 1|, it's at x = -1. These points divide the number line into different intervals where the expressions inside the absolute values are either positive or negative.Therefore, I can break down the problem into three cases:1. x < -12. -1 ≤ x ≤ 23. x > 2I'll analyze each case separately to see how the equation behaves.**Case 1: x < -1**In this interval, both x - 2 and x + 1 are negative because x is less than both -1 and 2. So, the absolute values will reverse their signs.So, |x - 2| becomes -(x - 2) = -x + 2, and |x + 1| becomes -(x + 1) = -x - 1.Adding them together: (-x + 2) + (-x - 1) = -2x + 1.So, the equation becomes: -2x + 1 = B.I can solve for x: -2x = B - 1 => x = (1 - B)/2.Now, since we're in the case where x < -1, we need to check if this solution satisfies x < -1.So, (1 - B)/2 < -1 => 1 - B < -2 => -B < -3 => B > 3.Therefore, if B > 3, there is a solution in this interval. But since we're looking for integer solutions, I need to see if (1 - B)/2 is an integer.But wait, B is given, but in the problem statement, it's not specified. Hmm, maybe I need to find the number of integer solutions regardless of B? Or is B a specific value? Wait, the problem says "there are C integers that satisfy the equation |x - 2| + |x + 1| = B", so I think B is given, but it's not specified here. Maybe I need to find C in terms of B? Or perhaps the problem is asking for the number of integer solutions in general?Wait, the original problem says: "Given that there are C integers that satisfy the equation |x - 2| + |x + 1| = B, find the value of C."Hmm, so it's given that there are C integer solutions, and I need to find what C is. So, maybe C is a specific number, regardless of B? Or perhaps C depends on B? Wait, the way it's phrased, it seems like C is a specific value, so maybe it's asking for the number of integer solutions in general.Wait, let me think again. The equation |x - 2| + |x + 1| = B. The left side is the sum of two absolute values, which is a function of x. Let me analyze this function.The function f(x) = |x - 2| + |x + 1| is a piecewise linear function. Let me find its minimum value because the number of solutions depends on the value of B relative to this minimum.To find the minimum, I can consider the function f(x). The sum of two absolute values is minimized when x is between the two points, which are -1 and 2. So, for x between -1 and 2, the function f(x) is constant.Wait, let me check that. Let's compute f(x) for x between -1 and 2.For x between -1 and 2, |x - 2| = 2 - x and |x + 1| = x + 1. So, f(x) = (2 - x) + (x + 1) = 3. So, f(x) is 3 for all x between -1 and 2.For x < -1, f(x) = |x - 2| + |x + 1| = (2 - x) + (-x - 1) = 2 - x - x - 1 = -2x + 1.For x > 2, f(x) = |x - 2| + |x + 1| = (x - 2) + (x + 1) = 2x - 1.So, the function f(x) has a V-shape with a minimum value of 3 between x = -1 and x = 2, and it increases linearly as x moves away from this interval.Therefore, the equation |x - 2| + |x + 1| = B will have:- No solutions if B < 3.- Infinitely many solutions if B = 3, specifically all x between -1 and 2.- Two solutions if B > 3: one to the left of -1 and one to the right of 2.But wait, the problem is asking for the number of integer solutions. So, let's consider each case.If B < 3: No solutions.If B = 3: All x between -1 and 2. Since we're looking for integer solutions, the integers in this interval are x = -1, 0, 1, 2. So, that's 4 integer solutions.If B > 3: Two solutions, one on each side of the interval. But these solutions may or may not be integers depending on B.Wait, but the problem says "there are C integers that satisfy the equation". So, it's given that there are C integers, and we need to find C.But depending on B, C can be 0, 4, or 2. But the problem doesn't specify B, so maybe it's asking for the maximum number of integer solutions possible? Or perhaps it's assuming B is such that there are C integer solutions, and we need to find C.Wait, maybe I need to consider that for B = 3, there are 4 integer solutions, and for B > 3, there are 2 integer solutions, and for B < 3, there are none. So, the possible values of C are 0, 2, or 4.But the problem says "there are C integers that satisfy the equation", so it's given that there are C integers, and we need to find C. So, maybe it's asking for the number of integer solutions in general, which is 4 when B = 3, and 2 when B > 3, and 0 otherwise.But the problem doesn't specify B, so perhaps it's asking for the number of integer solutions when B is such that there are solutions, which could be 4 or 2. Hmm, I'm a bit confused.Wait, maybe I need to consider that the problem is asking for the number of integer solutions regardless of B, but that doesn't make sense because the number of solutions depends on B.Alternatively, maybe the problem is asking for the number of integer solutions for all possible B, but that would be more complicated.Wait, perhaps the problem is just asking for the number of integer solutions when B is such that the equation has solutions, and we need to find the possible values of C. But the problem says "find the value of C", implying that C is a specific number.Wait, maybe I misread the problem. Let me check again."Given that there are C integers that satisfy the equation |x - 2| + |x + 1| = B, find the value of C."So, it's given that there are C integers, and we need to find C. So, perhaps the problem is implying that for some B, there are C integer solutions, and we need to find what C is. But without knowing B, it's impossible to determine C uniquely.Wait, maybe the problem is asking for the number of integer solutions in general, regardless of B, but that doesn't make sense because the number of solutions depends on B.Alternatively, perhaps the problem is asking for the number of integer solutions when B is such that the equation has solutions, and we need to find the possible values of C. But the problem says "find the value of C", which suggests a specific answer.Wait, maybe the problem is assuming that B is such that the equation has solutions, and we need to find the number of integer solutions. But without knowing B, it's unclear.Wait, perhaps the problem is asking for the number of integer solutions when B is such that the equation has exactly C solutions, and we need to find C. But that still doesn't make sense.Wait, maybe I need to consider that the problem is asking for the number of integer solutions for all possible B, but that would be an infinite number of solutions, which doesn't make sense.Wait, perhaps the problem is asking for the number of integer solutions when B is such that the equation has solutions, and we need to find the maximum number of integer solutions possible. In that case, the maximum number is 4, which occurs when B = 3.Alternatively, maybe the problem is asking for the number of integer solutions when B is such that the equation has exactly two solutions, which would be when B > 3, giving two integer solutions.But the problem doesn't specify, so I'm a bit stuck.Wait, maybe I need to consider that the problem is asking for the number of integer solutions in general, and the answer is 4 because when B = 3, there are 4 integer solutions, and that's the maximum.Alternatively, perhaps the problem is asking for the number of integer solutions when B is such that the equation has solutions, and the answer is 4 because that's the number of integer solutions when B = 3, which is the minimum value.Wait, I think I need to go back to the problem statement and make sure I understand it correctly."Given that there are C integers that satisfy the equation |x - 2| + |x + 1| = B, find the value of C."So, it's given that there are C integers, and we need to find C. So, perhaps the problem is implying that for some B, the equation has C integer solutions, and we need to find what C is.But without knowing B, it's impossible to determine C uniquely. Therefore, maybe the problem is asking for the number of integer solutions when B is such that the equation has solutions, and we need to find the possible values of C.Wait, but the problem says "find the value of C", which suggests a specific answer. So, perhaps the problem is assuming that B is such that the equation has solutions, and we need to find the number of integer solutions, which could be 4 or 2.Wait, maybe the problem is asking for the number of integer solutions when B is such that the equation has solutions, and the answer is 4 because that's the number of integer solutions when B = 3, which is the minimum value.Alternatively, perhaps the problem is asking for the number of integer solutions when B is such that the equation has exactly two solutions, which would be when B > 3, giving two integer solutions.But I'm still confused because the problem doesn't specify B.Wait, maybe I need to consider that the problem is asking for the number of integer solutions for all possible B, but that would be more complicated.Wait, perhaps the problem is asking for the number of integer solutions when B is such that the equation has solutions, and the answer is 4 because that's the number of integer solutions when B = 3, which is the minimum value.Alternatively, maybe the problem is asking for the number of integer solutions when B is such that the equation has exactly two solutions, which would be when B > 3, giving two integer solutions.But without knowing B, it's impossible to determine C uniquely.Wait, maybe the problem is asking for the number of integer solutions when B is such that the equation has solutions, and the answer is 4 because that's the number of integer solutions when B = 3, which is the minimum value.Alternatively, perhaps the problem is asking for the number of integer solutions when B is such that the equation has exactly two solutions, which would be when B > 3, giving two integer solutions.But the problem doesn't specify, so I'm stuck.Wait, maybe I need to consider that the problem is asking for the number of integer solutions in general, and the answer is 4 because when B = 3, there are 4 integer solutions, and that's the maximum.Alternatively, perhaps the problem is asking for the number of integer solutions when B is such that the equation has solutions, and the answer is 4 because that's the number of integer solutions when B = 3, which is the minimum value.Wait, I think I need to make a decision here. Given that the problem is asking for the value of C, and considering that when B = 3, there are 4 integer solutions, which is the maximum number of solutions, I think the answer is 4.So, C = 4.

answer：First, I need to determine the decimal representation of the fraction ( frac{8}{13} ). Performing the long division of 8 by 13, I find that it results in a repeating decimal: ( 0.overline{615384} ). This means the sequence "615384" repeats indefinitely.Next, I examine the repeating block "615384" to identify how many times the digit 3 appears. Upon reviewing the sequence, I notice that the digit 3 occurs exactly once within this 6-digit cycle.To calculate the probability of randomly selecting the digit 3 from this repeating sequence, I consider the ratio of the number of times 3 appears to the total number of digits in the repeating block. Since there is 1 occurrence of the digit 3 out of 6 total digits, the probability is ( frac{1}{6} ).

answer：Alright, so I have this problem about a sequence {a_n} where a_1 is 1, and each term is defined by the relation a_n = a_{n+1}^2 + 2a_{n+1}. There are two parts to this problem. Let me try to tackle them one by one.Starting with part (I): I need to prove that the sequence {log_2(a_n + 1)} is a geometric progression. Hmm, okay. So, first, I should recall what a geometric progression is—it's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio.Given that, I need to show that log_2(a_n + 1) divided by log_2(a_{n-1} + 1) is a constant ratio for all n. Alternatively, maybe I can find a relationship between log_2(a_{n+1} + 1) and log_2(a_n + 1).Looking at the given recurrence relation: a_n = a_{n+1}^2 + 2a_{n+1}. Maybe I can manipulate this equation to express a_n + 1 in terms of a_{n+1} + 1. Let me try that.So, starting with a_n = a_{n+1}^2 + 2a_{n+1}, if I add 1 to both sides, I get:a_n + 1 = a_{n+1}^2 + 2a_{n+1} + 1.Hmm, the right-hand side looks like a perfect square. Let me check:a_{n+1}^2 + 2a_{n+1} + 1 = (a_{n+1} + 1)^2.Yes, that's correct! So, we have:a_n + 1 = (a_{n+1} + 1)^2.Taking the logarithm base 2 of both sides:log_2(a_n + 1) = log_2((a_{n+1} + 1)^2).Using the logarithm power rule, log_b(x^k) = k*log_b(x), so:log_2(a_n + 1) = 2*log_2(a_{n+1} + 1).Hmm, so this gives a relationship between log_2(a_n + 1) and log_2(a_{n+1} + 1). Let me rearrange this:log_2(a_{n+1} + 1) = (1/2)*log_2(a_n + 1).Ah, so each term is half of the previous term. That means the sequence {log_2(a_n + 1)} is a geometric progression with common ratio 1/2.Let me verify this with the initial term. We know that a_1 = 1, so log_2(a_1 + 1) = log_2(2) = 1. Then, according to the recurrence, log_2(a_2 + 1) should be (1/2)*log_2(a_1 + 1) = 1/2. Let me check if that's true.From the original recurrence, a_1 = a_2^2 + 2a_2. Since a_1 is 1, we have:1 = a_2^2 + 2a_2.This is a quadratic equation: a_2^2 + 2a_2 - 1 = 0.Solving for a_2, we can use the quadratic formula:a_2 = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2*sqrt(2)] / 2 = -1 ± sqrt(2).Since the sequence is positive, a_2 must be positive, so a_2 = -1 + sqrt(2). Let me compute a_2 + 1:a_2 + 1 = (-1 + sqrt(2)) + 1 = sqrt(2).So, log_2(a_2 + 1) = log_2(sqrt(2)) = log_2(2^{1/2}) = 1/2. Perfect, that matches the geometric progression with ratio 1/2.So, part (I) seems to be proven. The key was to express a_n + 1 in terms of a_{n+1} + 1, recognize the perfect square, take logarithms, and then see the ratio.Moving on to part (II): Let b_n = n * log_2(a_n + 1), and let S_n be the sum of the first n terms of {b_n}. We need to prove that 1 ≤ S_n < 4.First, from part (I), we know that log_2(a_n + 1) is a geometric progression with first term 1 and common ratio 1/2. So, log_2(a_n + 1) = (1/2)^{n-1}.Therefore, b_n = n * (1/2)^{n-1}.So, S_n = sum_{k=1}^n k*(1/2)^{k-1}.I need to compute this sum and show that it's always between 1 and 4.Let me write out the first few terms to get a sense:When n=1: S_1 = 1*(1/2)^0 = 1*1 = 1.When n=2: S_2 = 1*(1/2)^0 + 2*(1/2)^1 = 1 + 2*(1/2) = 1 + 1 = 2.When n=3: S_3 = 1 + 1 + 3*(1/2)^2 = 1 + 1 + 3*(1/4) = 2 + 3/4 = 2.75.When n=4: S_4 = 1 + 1 + 3/4 + 4*(1/2)^3 = 1 + 1 + 3/4 + 4*(1/8) = 2 + 3/4 + 0.5 = 3.25.Hmm, it's increasing, but each term is getting smaller because of the (1/2)^{k-1} factor.I need to find a general expression for S_n.Let me recall that the sum S = sum_{k=1}^∞ k*x^{k-1} is a known series. It's the derivative of the geometric series.Indeed, the sum from k=1 to infinity of x^k is x/(1 - x) for |x| < 1. Taking derivative with respect to x:sum_{k=1}^∞ k*x^{k-1} = 1/(1 - x)^2.So, in our case, x = 1/2, so the infinite sum would be 1/(1 - 1/2)^2 = 1/(1/2)^2 = 4.Therefore, as n approaches infinity, S_n approaches 4. But since we're summing up to n, S_n is less than 4 for all finite n.Also, since each term b_k is positive, S_n is increasing. The first term is 1, so S_n is at least 1.Therefore, 1 ≤ S_n < 4.But let me verify this more carefully, perhaps by induction or by using the formula for finite sums.The finite sum S_n = sum_{k=1}^n k*(1/2)^{k-1}.To find a closed-form expression, let's denote S = sum_{k=1}^n k*r^{k-1}, where r = 1/2.We can use the formula for the finite sum:S = (1 - (n+1)*r^n + n*r^{n+1}) / (1 - r)^2.Plugging in r = 1/2:S = (1 - (n+1)*(1/2)^n + n*(1/2)^{n+1}) / (1 - 1/2)^2.Simplify denominator: (1 - 1/2)^2 = (1/2)^2 = 1/4.So, S = [1 - (n+1)*(1/2)^n + n*(1/2)^{n+1}] / (1/4) = 4*[1 - (n+1)*(1/2)^n + n*(1/2)^{n+1}].Simplify the numerator:1 - (n+1)*(1/2)^n + n*(1/2)^{n+1} = 1 - (n+1)*(1/2)^n + (n/2)*(1/2)^n = 1 - (n+1 - n/2)*(1/2)^n = 1 - (n/2 + 1)*(1/2)^n.Therefore, S = 4*[1 - (n/2 + 1)*(1/2)^n] = 4 - 4*(n/2 + 1)*(1/2)^n.Simplify 4*(n/2 + 1)*(1/2)^n:4*(n/2 + 1) = 2n + 4.So, S = 4 - (2n + 4)*(1/2)^n.Which can be written as S = 4 - (n + 2)*(1/2)^{n -1}.Wait, let me check that:Wait, 4*(n/2 +1)*(1/2)^n = 4*( (n + 2)/2 )*(1/2)^n = 2*(n + 2)*(1/2)^n = (n + 2)*(1/2)^{n -1}.Yes, because 2*(1/2)^n = (1/2)^{n -1}.So, S = 4 - (n + 2)*(1/2)^{n -1}.Therefore, S_n = 4 - (n + 2)*(1/2)^{n -1}.Since (n + 2)*(1/2)^{n -1} is positive for all n ≥ 1, it follows that S_n < 4.Also, since each term b_k is positive, S_n is increasing. The first term S_1 = 1, so S_n ≥ 1 for all n.Therefore, 1 ≤ S_n < 4.Let me test this formula with n=1,2,3,4 to see if it holds.For n=1:S_1 = 4 - (1 + 2)*(1/2)^{0} = 4 - 3*1 = 1. Correct.For n=2:S_2 = 4 - (2 + 2)*(1/2)^{1} = 4 - 4*(1/2) = 4 - 2 = 2. Correct.For n=3:S_3 = 4 - (3 + 2)*(1/2)^{2} = 4 - 5*(1/4) = 4 - 1.25 = 2.75. Correct.For n=4:S_4 = 4 - (4 + 2)*(1/2)^{3} = 4 - 6*(1/8) = 4 - 0.75 = 3.25. Correct.So, the formula seems to hold. Therefore, S_n is indeed 4 minus a positive term, so it's less than 4, and since each term is positive, it's at least 1.Hence, both parts are proven.**Final Answer**(I) The sequence is a geometric progression with common ratio boxed{frac{1}{2}}.(II) The sum satisfies boxed{1 leq S_n < 4}.

answer：To solve the problem, I'll start by defining the unknown number as ( x ).The problem states that adding 12 to 60 and then dividing by 3 is equal to the number subtracted by 12 and then multiplied by 3. This can be written as the equation:[frac{60 + 12}{3} = (x - 12) times 3]First, I'll simplify the left side of the equation:[frac{72}{3} = (x - 12) times 3][24 = (x - 12) times 3]Next, I'll solve for ( x ) by expanding the right side of the equation:[24 = 3x - 36]To isolate ( x ), I'll add 36 to both sides of the equation:[24 + 36 = 3x][60 = 3x]Finally, I'll divide both sides by 3 to find the value of ( x ):[frac{60}{3} = x][20 = x]Therefore, the number we are looking for is 20.

answer：First, I need to calculate the final price of the smartphone at the first store, which offers a single discount of 25%. - The original price of the smartphone is 800.- A 25% discount means the customer pays 75% of the original price.- Calculating 75% of 800: [ 800 times 0.75 = 600 ]- So, the final price at the first store is 600.Next, I need to calculate the final price at the second store, which offers two successive discounts of 20% and 10%.- Starting with the original price of 800, the first discount is 20%.- A 20% discount means the customer pays 80% of the original price.- Calculating 80% of 800: [ 800 times 0.80 = 640 ]- The price after the first discount is 640.- Now, applying the second discount of 10% on the new price of 640.- A 10% discount means the customer pays 90% of the current price.- Calculating 90% of 640: [ 640 times 0.90 = 576 ]- So, the final price at the second store is 576.Finally, I need to find the difference in final prices between the two stores.- The final price at the first store is 600.- The final price at the second store is 576.- Calculating the difference: [ 600 - 576 = 24 ]- The difference in final prices is 24.

answer：Okay, so I need to find the sum of two things: the greatest common divisor (GCD) of 42 and 56, and the least common multiple (LCM) of 24 and 18. Hmm, let's break this down step by step.First, I'll tackle the GCD of 42 and 56. I remember that the GCD is the largest number that divides both numbers without leaving a remainder. One way to find the GCD is by listing out all the factors of each number and then finding the largest common one. Let's try that.For 42, the factors are: 1, 2, 3, 6, 7, 14, 21, 42.For 56, the factors are: 1, 2, 4, 7, 8, 14, 28, 56.Now, looking at both lists, the common factors are 1, 2, 7, and 14. Out of these, the largest one is 14. So, the GCD of 42 and 56 is 14.Next, I need to find the LCM of 24 and 18. The LCM is the smallest number that both 24 and 18 can divide into without leaving a remainder. I think one method to find the LCM is by listing the multiples of each number until I find the smallest common one.Multiples of 24: 24, 48, 72, 96, 120, ...Multiples of 18: 18, 36, 54, 72, 90, 108, 126, ...Looking at these lists, the first common multiple is 72. So, the LCM of 24 and 18 is 72.Now, I need to add the GCD and the LCM together. That would be 14 (GCD) + 72 (LCM) = 86.Wait, let me double-check to make sure I didn't make any mistakes. For the GCD, 14 divides both 42 and 56 evenly, and there's no larger number that does that. For the LCM, 72 is indeed the smallest number that both 24 and 18 can divide into without a remainder. So, adding them together gives me 86. I think that's correct!