💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand it step by step. The problem says:"An arithmetic sequence consists of 150 numbers that are each at least 20 and at most 90. The sum of the numbers is 9,000. Let L' be the least possible value of the 40th term and let G' be the greatest possible value of the 40th term. What is the value of G' - L'?"Alright, so first, let's recall what an arithmetic sequence is. It's a sequence where each term after the first is obtained by adding a constant difference. So, if I denote the first term as a1 and the common difference as d, then the nth term is a1 + (n-1)d.Given that there are 150 terms, each term is between 20 and 90, and the total sum is 9,000. I need to find the maximum and minimum possible values of the 40th term and then find their difference.First, let's think about the sum of the arithmetic sequence. The formula for the sum of an arithmetic sequence is S = n/2 * (2a1 + (n-1)d). Here, n is 150, and S is 9,000.So, plugging in the values:9,000 = (150)/2 * (2a1 + 149d)Simplify that:9,000 = 75 * (2a1 + 149d)Divide both sides by 75:120 = 2a1 + 149dSo, 2a1 + 149d = 120.Let me write that as equation (1):2a1 + 149d = 120.Now, I also know that each term is at least 20 and at most 90. So, the first term a1 must be >=20, and the last term a150 must be <=90.Wait, actually, the first term could be as low as 20, but the last term could be as high as 90. But in an arithmetic sequence, the terms increase (or decrease) by a constant difference. So, depending on whether d is positive or negative, the sequence could be increasing or decreasing.But in this case, since we have 150 terms, each between 20 and 90, the sequence could be increasing or decreasing. Hmm, but the sum is fixed at 9,000, so the average term is 9,000 / 150 = 60.So, the average term is 60. In an arithmetic sequence, the average is equal to the middle term. Since there are 150 terms, which is even, the average would be the average of the 75th and 76th terms. But for simplicity, I can think of the sequence as symmetrically distributed around 60.Wait, maybe it's better to think of the sequence as centered around 60. So, the first term is 60 - 149d/2, and the last term is 60 + 149d/2. Hmm, is that right?Wait, let me think again. The nth term of an arithmetic sequence is a1 + (n-1)d. So, the 1st term is a1, the 150th term is a1 + 149d.If the average term is 60, then the average of the first and last term is 60. So, (a1 + a150)/2 = 60.Which implies that a1 + a150 = 120.But a150 = a1 + 149d, so:a1 + (a1 + 149d) = 120Which simplifies to 2a1 + 149d = 120, which is the same as equation (1). So, that checks out.So, we have 2a1 + 149d = 120.Now, the constraints are that each term is between 20 and 90. So, the first term a1 >=20, and the last term a150 <=90.But actually, not just the first and last terms, but every term in between must also satisfy 20 <= a_n <=90.So, for all n from 1 to 150, 20 <= a1 + (n-1)d <=90.So, to ensure that all terms are within 20 and 90, we need to make sure that the minimum term is >=20 and the maximum term is <=90.In an arithmetic sequence, the minimum term is either the first term or the last term, depending on whether the sequence is increasing or decreasing.Similarly, the maximum term is either the last term or the first term, depending on the direction.So, if d is positive, the sequence is increasing, so the first term is the minimum, and the last term is the maximum.If d is negative, the sequence is decreasing, so the first term is the maximum, and the last term is the minimum.Therefore, in either case, we have:If d > 0:a1 >=20a150 <=90If d < 0:a1 <=90a150 >=20But since we don't know the sign of d, we have to consider both possibilities.But perhaps it's better to express the constraints in terms of a1 and d.So, for all n, 20 <= a1 + (n-1)d <=90.Which can be rewritten as:20 - a1 <= (n-1)d <=90 - a1But since this must hold for all n from 1 to 150, the most restrictive cases are when n=1 and n=150.So, for n=1:20 <= a1 <=90For n=150:20 <= a1 + 149d <=90So, combining these, we have:20 <= a1 <=90and20 <= a1 + 149d <=90So, from the first inequality, a1 is between 20 and 90.From the second inequality, a1 + 149d is between 20 and 90.But since we also have the equation 2a1 + 149d = 120, we can use that to relate a1 and d.Let me solve equation (1) for a1:2a1 = 120 - 149dSo,a1 = (120 - 149d)/2So, a1 = 60 - (149d)/2Now, since a1 must be >=20 and <=90, let's plug this into the inequalities.First, a1 >=20:60 - (149d)/2 >=20Subtract 60:- (149d)/2 >= -40Multiply both sides by -2 (remember to reverse inequality):149d <=80So,d <=80/149Similarly, a1 <=90:60 - (149d)/2 <=90Subtract 60:- (149d)/2 <=30Multiply both sides by -2 (reverse inequality):149d >= -60So,d >= -60/149So, from a1's constraints, we have:-60/149 <= d <=80/149Now, let's look at the other inequality involving a150:a1 + 149d >=20But a1 +149d = (60 - (149d)/2) +149d =60 + (149d)/2So,60 + (149d)/2 >=20Subtract 60:(149d)/2 >= -40Multiply both sides by 2:149d >= -80So,d >= -80/149Similarly, a1 +149d <=90:60 + (149d)/2 <=90Subtract 60:(149d)/2 <=30Multiply both sides by 2:149d <=60So,d <=60/149So, combining all these constraints:From a1 >=20: d <=80/149From a1 <=90: d >=-60/149From a150 >=20: d >=-80/149From a150 <=90: d <=60/149So, the most restrictive constraints are:d >= max(-60/149, -80/149) = -60/149 (since -60/149 > -80/149)andd <= min(80/149,60/149) =60/149So, overall, d must satisfy:-60/149 <=d <=60/149Wait, but earlier from a1 <=90, we had d >=-60/149, and from a150 >=20, we had d >=-80/149. So, the lower bound is the higher of these two, which is -60/149.Similarly, the upper bound is the lower of 80/149 and 60/149, which is 60/149.So, d is between -60/149 and 60/149.But wait, let me double-check.From a1 >=20: d <=80/149From a1 <=90: d >=-60/149From a150 >=20: d >=-80/149From a150 <=90: d <=60/149So, combining these:d must be >= max(-60/149, -80/149) = -60/149and d <= min(80/149,60/149)=60/149So, yes, d is between -60/149 and 60/149.Wait, but let me think again. If d is positive, the sequence is increasing, so a1 is the minimum, a150 is the maximum.If d is negative, the sequence is decreasing, so a1 is the maximum, a150 is the minimum.So, in both cases, we have:If d >0:a1 >=20a150 <=90If d <0:a1 <=90a150 >=20But in our earlier analysis, we considered both a1 and a150 constraints, leading to d between -60/149 and 60/149.But let's see if that makes sense.Wait, if d is positive, then a1 >=20 and a150 <=90.From a1 >=20: d <=80/149From a150 <=90: d <=60/149So, for d positive, the upper bound is 60/149.Similarly, for d negative, a1 <=90 and a150 >=20.From a1 <=90: d >=-60/149From a150 >=20: d >=-80/149So, the lower bound is -60/149.So, overall, d is between -60/149 and 60/149.Okay, that seems consistent.Now, our goal is to find the 40th term, which is a1 + 39d.We need to find the minimum and maximum possible values of this term, given the constraints on d.So, let's express the 40th term in terms of d.We have a1 =60 - (149d)/2So, a40 = a1 +39d =60 - (149d)/2 +39dSimplify:60 + (39d - (149d)/2)Convert 39d to halves: 78d/2 -149d/2 = (-71d)/2So, a40 =60 - (71d)/2Wait, let me check that again.a1 =60 - (149d)/2a40 = a1 +39d =60 - (149d)/2 +39dConvert 39d to halves: 39d =78d/2So,60 - (149d)/2 +78d/2 =60 + (-149d +78d)/2 =60 + (-71d)/2So, a40 =60 - (71d)/2Yes, that's correct.So, a40 =60 - (71d)/2Now, we need to find the minimum and maximum of a40 given that d is between -60/149 and 60/149.So, let's analyze how a40 changes with d.Since a40 =60 - (71d)/2, it's a linear function of d.The coefficient of d is negative (-71/2), which means that as d increases, a40 decreases, and as d decreases, a40 increases.Therefore, to find the minimum value of a40, we need to maximize d.To find the maximum value of a40, we need to minimize d.Given that d is between -60/149 and 60/149.So,Maximum d =60/149Minimum d =-60/149Therefore,Minimum a40 occurs when d is maximum:a40_min =60 - (71*(60/149))/2Similarly,Maximum a40 occurs when d is minimum:a40_max =60 - (71*(-60/149))/2Let me compute these.First, compute a40_min:a40_min =60 - (71*(60/149))/2Compute 71*60: 71*60=4260So,a40_min =60 - (4260/149)/2Simplify:4260/149 = let's compute that.149*28=41724260 -4172=88So, 4260/149=28 +88/149=28.5906 approximately.But let's keep it as a fraction.So,4260/149 divided by 2 is 2130/149So,a40_min =60 -2130/149Convert 60 to 149 denominator:60=60*149/149=8840/149So,a40_min=8840/149 -2130/149=(8840-2130)/149=6710/149Similarly, compute a40_max:a40_max=60 - (71*(-60/149))/2=60 + (71*60)/(2*149)=60 +2130/149Again, 2130/149=14.295 approx.But let's compute 60 +2130/149.Convert 60 to 149 denominator:60=8840/149So,a40_max=8840/149 +2130/149=(8840+2130)/149=10970/149Now, we need to find G' - L', which is a40_max -a40_min.So,G' - L'= (10970/149) - (6710/149)= (10970 -6710)/149=4260/149Simplify 4260/149.Let me compute 149*28=41724260-4172=88So, 4260=149*28 +88So, 4260/149=28 +88/149But 88 and 149 have a common factor? Let's see.149 is a prime number, I think. Let me check: 149 divided by 2,3,5,7,11.149/2=74.5149/3≈49.666149/5=29.8149/7≈21.285149/11≈13.545So, no, 149 is prime.So, 88/149 cannot be simplified.So, 4260/149=28 +88/149=28.5906 approximately.But the question asks for the value of G' - L', so it's 4260/149.But let me check my calculations again to make sure.Wait, a40 =60 - (71d)/2So, when d is maximum (60/149), a40 is minimum:a40_min=60 - (71*(60/149))/2=60 - (4260/149)/2=60 -2130/149Similarly, when d is minimum (-60/149), a40 is maximum:a40_max=60 - (71*(-60/149))/2=60 +2130/149So, difference is (60 +2130/149) - (60 -2130/149)=4260/149Yes, that's correct.So, the difference G' - L' is 4260/149.But let me see if this can be simplified or if I made a mistake in the calculation.Wait, 4260 divided by 149.Let me compute 149*28=41724260-4172=88So, 4260=149*28 +88So, 4260/149=28 +88/149So, as a mixed number, it's 28 and 88/149, but as an improper fraction, it's 4260/149.But perhaps the problem expects it as a reduced fraction or a decimal.But since 149 is prime, and 88 and 149 have no common factors, 4260/149 is the simplest form.Wait, but let me check 4260 and 149.149*28=41724260-4172=88So, 4260=149*28 +88So, 4260/149=28 +88/149So, yes, that's correct.Alternatively, 4260 divided by 149:149*28=41724260-4172=88So, 4260/149=28 +88/149So, as a decimal, 88/149≈0.5906So, total≈28.5906But the problem doesn't specify the form, so probably as a fraction.But let me check if 4260 and 149 have any common factors.149 is prime, so let's check if 149 divides 4260.4260 divided by 149:149*28=41724260-4172=88So, no, 149 doesn't divide 4260 exactly, so 4260/149 is the simplest form.Wait, but 4260 and 149: 4260=4260, 149 is prime.So, 4260=2^2 *3 *5 *71Wait, 4260 divided by 2=21302130 divided by 2=10651065 divided by 5=213213 divided by 3=71So, 4260=2^2 *3 *5 *71149 is prime, and 149 doesn't divide into 4260, so yes, 4260/149 is the simplest form.Therefore, G' - L'=4260/149.But wait, in the initial problem, the user had a different approach, where they considered the sequence centered around 60, with terms like 60 -149d, 60 -147d, etc.Wait, let me see.In the initial problem, the user wrote:"The sequence can be represented as:60 - 149d, 60 - 147d, dots, 60 - d, 60 + d, dots, 60 + 147d, 60 + 149d."Wait, that seems incorrect because in an arithmetic sequence, the difference between consecutive terms is constant. So, if the first term is 60 -149d, the next term should be 60 -149d +d=60 -148d, then 60 -147d, etc.Wait, but the user wrote 60 -149d, 60 -147d, which skips 148d. So, that seems like a mistake.Wait, perhaps the user intended to write 60 -149d, 60 -148d, 60 -147d,...,60 -d,60,60 +d,...,60 +147d,60 +148d,60 +149d.But in that case, the common difference would be d, but the first term is 60 -149d, and the last term is 60 +149d.So, the number of terms would be (149d +149d)/d +1=298/d +1, which doesn't make sense because we have 150 terms.Wait, perhaps the user made a mistake in representing the sequence.Alternatively, perhaps they considered the middle term as 60, and the terms are symmetrically distributed around 60.But in that case, for 150 terms, the middle would be between 75th and 76th terms, both equal to 60.But in an arithmetic sequence, the terms are equally spaced, so the 75th term is 60 -74d, and the 76th term is 60 -73d, but if the average is 60, then (a75 +a76)/2=60.But perhaps the user tried to represent the sequence as centered around 60, but their representation seems off.But regardless, in my approach, I considered the general case, found a1 in terms of d, expressed the 40th term, and found the difference.So, I think my approach is correct, leading to G' - L'=4260/149.But let me check if 4260/149 can be simplified.As I did earlier, 4260=2^2 *3 *5 *71149 is prime, and 149 doesn't divide into 4260, so yes, it's in simplest terms.Alternatively, 4260 divided by 149 is 28.5906 approximately.But since the problem is likely expecting an exact value, 4260/149 is the answer.Wait, but let me check my earlier steps again to make sure I didn't make a mistake.We had:a40=60 - (71d)/2Then, d ranges from -60/149 to60/149.So, when d=60/149, a40=60 - (71*(60/149))/2=60 - (4260/149)/2=60 -2130/149Similarly, when d=-60/149, a40=60 - (71*(-60/149))/2=60 +2130/149So, the difference is 4260/149.Yes, that seems correct.Alternatively, perhaps I made a mistake in expressing a40.Wait, let's rederive a40.We have a1=60 - (149d)/2So, a40=a1 +39d=60 - (149d)/2 +39dConvert 39d to halves:39d=78d/2So,60 - (149d)/2 +78d/2=60 + (-149d +78d)/2=60 + (-71d)/2Yes, that's correct.So, a40=60 - (71d)/2So, when d is maximum (60/149), a40 is minimum.When d is minimum (-60/149), a40 is maximum.So, the difference is 4260/149.Therefore, the answer is 4260/149.But let me check if 4260 divided by 149 is indeed 28.5906.149*28=41724172+88=4260Yes, so 4260/149=28 +88/149≈28.5906But since the problem is about an arithmetic sequence with 150 terms, each between 20 and90, sum 9000, and finding the difference between the maximum and minimum 40th term, the answer is 4260/149.But wait, in the initial problem, the user had a different approach, leading to 6660/149.Wait, let me see.In the initial problem, the user wrote:"The sequence can be represented as:60 - 149d, 60 - 147d, dots, 60 - d, 60 + d, dots, 60 + 147d, 60 + 149d."But this seems incorrect because the common difference would be 2d, not d, since from 60 -149d to 60 -147d is an increase of 2d.So, perhaps the user considered the common difference as 2d, which would make the sequence have terms spaced by 2d, but that would mean the number of terms is different.Wait, in that case, the number of terms would be (149d +149d)/2d +1=149 +1=150 terms.Ah, so perhaps the user considered the common difference as 2d, so that the sequence is 60 -149d, 60 -147d,...,60 +149d, with a common difference of 2d.In that case, the number of terms is 150, since from -149d to +149d with step 2d gives (149d - (-149d))/2d +1= (298d)/2d +1=149 +1=150 terms.So, in that case, the common difference is 2d, and the first term is 60 -149d, last term is60 +149d.Then, the sum is 150*(first term +last term)/2=150*(120)/2=150*60=9000, which matches.So, in this representation, the common difference is 2d, and the nth term is60 -149d + (n-1)*2d.So, the 40th term would be60 -149d + (40-1)*2d=60 -149d +78d=60 -71d.Wait, so in this case, the 40th term is60 -71d.Wait, but in my earlier approach, I had a40=60 - (71d)/2.So, there's a discrepancy here.Wait, perhaps the user considered the common difference as 2d, so d in their representation is twice the d in my representation.So, in their terms, the common difference is 2d, so their d is my 2d.Therefore, their 40th term is60 -71d, where d is their common difference, which is 2d in my terms.So, in their terms, d is the common difference, which is 2d in my terms.So, in their terms, the constraints would be:From the first term:60 -149d >=20 =>d <=40/149From the last term:60 +149d <=90 =>d <=30/149So, d <=30/149Similarly, for the first term:60 -149d >=20 =>d <=40/149And for the last term:60 +149d <=90 =>d <=30/149So, d <=30/149Similarly, for the first term as maximum:60 -149d <=90 =>-149d <=30 =>d >=-30/149And for the last term as minimum:60 +149d >=20 =>149d >=-40 =>d >=-40/149So, d >=-40/149But since d is the common difference, which can be positive or negative.But in their approach, they considered the maximum d as30/149, and minimum d as-40/149.Wait, but in their solution, they wrote:"From the constraints 60 - 149d >=20 and 60 +149d <=90, we solve:60 -149d >=20 =>d <=40/14960 +149d <=90 =>d <=30/149Thus, the maximum value of |d| is30/149."Wait, but that's not correct because d can be negative as well.Wait, in their approach, they considered d as the common difference, which can be positive or negative.But in their constraints, they only considered d <=40/149 and d <=30/149, so d <=30/149.But they didn't consider the lower bounds.Wait, perhaps they assumed d is positive, but in reality, d can be negative as well.So, perhaps their approach is incomplete.But in any case, in their solution, they found that the 40th term is60 -111d, which seems incorrect because in their representation, the 40th term is60 -71d.Wait, let me check.In their representation, the nth term is60 -149d + (n-1)*2d.So, for n=40:a40=60 -149d +39*2d=60 -149d +78d=60 -71d.So, a40=60 -71d.But in their solution, they wrote:"The 40th term is60 -111d."Wait, that's incorrect.Wait, perhaps they made a mistake in calculating the 40th term.Wait, let's see.In their representation, the sequence is:60 -149d, 60 -147d, ...,60 -d,60 +d,...,60 +147d,60 +149d.So, each term increases by 2d.So, the first term is60 -149d, the second term is60 -147d, which is an increase of2d.So, the nth term is60 -149d + (n-1)*2d.So, for n=40:a40=60 -149d + (40-1)*2d=60 -149d +78d=60 -71d.So, a40=60 -71d.But in their solution, they wrote:"The 40th term is60 -111d."Which is incorrect.Wait, perhaps they considered the 40th term as60 - (149 - (40-1)*2)d=60 - (149 -78)d=60 -71d.Wait, that's the same as above.So, perhaps they made a mistake in their solution.In their solution, they wrote:"The 40th term is60 -111d."But according to the correct calculation, it's60 -71d.So, perhaps they made a mistake in their calculation.Therefore, their final answer of6660/149 is incorrect.In my approach, I correctly derived a40=60 - (71d)/2, considering the common difference as d, not 2d.So, in my approach, the difference G' - L' is4260/149.But let me check again.Wait, in my approach, I considered the common difference as d, so the nth term is a1 + (n-1)d.In the user's approach, they considered the common difference as2d, so the nth term is60 -149d + (n-1)*2d.So, in their terms, the common difference is2d, which is different from my d.Therefore, in their solution, when they wrote:"The 40th term is60 -111d."That's incorrect because, as we saw, it should be60 -71d.But in their solution, they wrote:"The 40th term is60 -111d."Which suggests that they thought the 40th term is60 -111d, which is not correct.So, their solution is flawed.Therefore, my approach is correct, leading to G' - L'=4260/149.But let me confirm once more.In my approach:a1=60 - (149d)/2a40=60 - (71d)/2d ranges from-60/149 to60/149So, a40 ranges from60 - (71*(60/149))/2=60 -2130/149= (60*149 -2130)/149=(8840 -2130)/149=6710/149To60 - (71*(-60/149))/2=60 +2130/149=(60*149 +2130)/149=(8840 +2130)/149=10970/149So, the difference is10970/149 -6710/149=4260/149.Yes, that's correct.Therefore, the answer is4260/149.But let me check if 4260/149 can be simplified.As I did earlier, 4260 divided by149 is28 with a remainder of88, so4260=149*28 +88.So,4260/149=28 +88/149.Since88 and149 have no common factors, this is the simplest form.Therefore, the value ofG' - L' is4260/149.But wait, in the initial problem, the user had a different answer,6660/149.So, perhaps I made a mistake in my approach.Wait, let me check.In my approach, I considered the common difference as d, so the nth term is a1 + (n-1)d.In the user's approach, they considered the common difference as2d, so the nth term is60 -149d + (n-1)*2d.So, in their terms, the 40th term is60 -71d.But in their solution, they wrote:"The 40th term is60 -111d."Which is incorrect.So, perhaps they made a mistake in their calculation.But in my approach, I correctly derived a40=60 - (71d)/2.Wait, but in my approach, I considered the common difference as d, so the nth term is a1 + (n-1)d.But in the user's approach, they considered the common difference as2d, so their nth term is60 -149d + (n-1)*2d.So, in their terms, the 40th term is60 -71d.But in their solution, they wrote:"The 40th term is60 -111d."Which is incorrect.So, their solution is wrong.Therefore, my approach is correct, leading to G' - L'=4260/149.But let me check if 4260/149 is equal to6660/149 divided by something.Wait, 6660/149 is approximately44.7, while4260/149≈28.59.So, they are different.Therefore, my answer is4260/149.But let me check if I made a mistake in expressing a40.Wait, in my approach, I had a1=60 - (149d)/2So, a40=a1 +39d=60 - (149d)/2 +39d=60 - (149d -78d)/2=60 -71d/2.Yes, that's correct.So, a40=60 - (71d)/2.Therefore, when d=60/149, a40=60 - (71*(60/149))/2=60 - (4260/149)/2=60 -2130/149.Similarly, when d=-60/149, a40=60 - (71*(-60/149))/2=60 +2130/149.So, the difference is4260/149.Yes, that's correct.Therefore, the answer is4260/149.But let me check if this is the same as the user's answer.The user's answer was6660/149, which is approximately44.7.But in my approach, the difference is4260/149≈28.59.So, they are different.Therefore, I think my approach is correct, and the user made a mistake in their solution.Therefore, the correct answer is4260/149.

💡To find the range of the function ( y = x^2 + x ) for the interval ( -1 leq x leq 3 ), we can follow these steps:1. **Complete the Square:** Rewrite the quadratic function in vertex form by completing the square. [ y = x^2 + x = left(x + frac{1}{2}right)^2 - frac{1}{4} ] This shows that the vertex of the parabola is at ( left(-frac{1}{2}, -frac{1}{4}right) ).2. **Determine the Behavior of the Function:** Since the coefficient of ( x^2 ) is positive, the parabola opens upwards. This means the function has a minimum value at the vertex and increases as we move away from the vertex in both directions.3. **Evaluate the Function at Critical Points:** - **Vertex:** At ( x = -frac{1}{2} ), the function reaches its minimum value: [ y = left(-frac{1}{2}right)^2 + left(-frac{1}{2}right) = frac{1}{4} - frac{1}{2} = -frac{1}{4} ] - **Endpoints of the Interval:** - At ( x = -1 ): [ y = (-1)^2 + (-1) = 1 - 1 = 0 ] - At ( x = 3 ): [ y = 3^2 + 3 = 9 + 3 = 12 ]4. **Determine the Range:** The function attains its minimum value at the vertex ( x = -frac{1}{2} ) and its maximum value at the endpoint ( x = 3 ). Therefore, the range of the function over the interval ( -1 leq x leq 3 ) is: [ left[ -frac{1}{4}, 12 right] ]Thus, the correct answer is option B.

💡Okay, so I need to find the maximum and minimum values of the function y = 2 sin(3x + π/3). Hmm, I remember that the sine function usually oscillates between -1 and 1. So, if I have 2 sin(something), the maximum should be 2 and the minimum should be -2, right? That seems straightforward.But wait, the function inside the sine is 3x + π/3. Does that affect the maximum and minimum values? I think the amplitude is still 2 because it's multiplied by 2 outside the sine function. The 3x inside just affects the period and the phase shift, not the amplitude. So, yeah, the maximum value is 2 and the minimum is -2. That part seems clear.Now, the tricky part is finding the set of x values that achieve these maximum and minimum values. Let me think about how to approach this. For the maximum value, which is 2, the sine function must be equal to 1. So, I can set up the equation:2 sin(3x + π/3) = 2Dividing both sides by 2 gives:sin(3x + π/3) = 1I know that sin(θ) = 1 when θ = π/2 + 2πk, where k is any integer. So, substituting back:3x + π/3 = π/2 + 2πkNow, I need to solve for x. Let me subtract π/3 from both sides:3x = π/2 - π/3 + 2πkTo combine π/2 and -π/3, I need a common denominator. The common denominator for 2 and 3 is 6. So, π/2 is 3π/6 and -π/3 is -2π/6. Adding them together:3π/6 - 2π/6 = π/6So now, we have:3x = π/6 + 2πkDividing both sides by 3:x = π/18 + (2πk)/3So, the x values where the function reaches its maximum are x = π/18 + (2πk)/3 for any integer k.Now, let's do the same for the minimum value, which is -2. So, we set up the equation:2 sin(3x + π/3) = -2Dividing both sides by 2:sin(3x + π/3) = -1I remember that sin(θ) = -1 when θ = 3π/2 + 2πk, where k is any integer. So, substituting back:3x + π/3 = 3π/2 + 2πkAgain, solving for x. Subtract π/3 from both sides:3x = 3π/2 - π/3 + 2πkLet me find a common denominator for 3π/2 and -π/3. The common denominator is 6. So, 3π/2 is 9π/6 and -π/3 is -2π/6. Adding them together:9π/6 - 2π/6 = 7π/6So now, we have:3x = 7π/6 + 2πkDividing both sides by 3:x = 7π/18 + (2πk)/3Wait, hold on. Let me double-check that. If 3x = 7π/6 + 2πk, then dividing by 3 gives x = (7π/6)/3 + (2πk)/3, which is 7π/18 + (2πk)/3. Yeah, that seems correct.But wait, I thought when sin(θ) = -1, θ is 3π/2 + 2πk. So, 3x + π/3 = 3π/2 + 2πk. Then, solving for x:3x = 3π/2 - π/3 + 2πk3π/2 is 9π/6, and π/3 is 2π/6, so 9π/6 - 2π/6 = 7π/6. So, 3x = 7π/6 + 2πk, so x = 7π/18 + (2πk)/3. Hmm, that seems correct.But wait, I remember sometimes when solving trigonometric equations, you can have different expressions depending on how you manipulate the equation. Let me see if I can express this differently.Alternatively, 3x + π/3 = 3π/2 + 2πkSubtract π/3:3x = 3π/2 - π/3 + 2πkWhich is the same as 3x = (9π/6 - 2π/6) + 2πk = 7π/6 + 2πkSo, x = 7π/18 + (2πk)/3Alternatively, I could factor out π/18:x = π/18 (7 + 12k)But that might not be necessary. So, the x values where the function reaches its minimum are x = 7π/18 + (2πk)/3 for any integer k.Wait, but let me check if this makes sense. Let me plug in k = 0 for the maximum:x = π/18 ≈ 0.1745 radiansPlugging into the original function:y = 2 sin(3*(π/18) + π/3) = 2 sin(π/6 + π/3) = 2 sin(π/2) = 2*1 = 2. Correct.For the minimum, x = 7π/18 ≈ 1.2217 radiansPlugging into the original function:y = 2 sin(3*(7π/18) + π/3) = 2 sin(7π/6 + π/3) = 2 sin(7π/6 + 2π/6) = 2 sin(9π/6) = 2 sin(3π/2) = 2*(-1) = -2. Correct.Good, so the calculations seem to check out.So, to summarize:- The maximum value of y is 2, achieved when x = π/18 + (2πk)/3 for any integer k.- The minimum value of y is -2, achieved when x = 7π/18 + (2πk)/3 for any integer k.Alternatively, I can write the solutions as:For maximum:x = (π/18) + (2π/3)kFor minimum:x = (7π/18) + (2π/3)kWhere k is any integer.I think that's it. I don't see any mistakes in my reasoning. I double-checked by plugging in specific values of k and ensuring that the function indeed reaches the maximum and minimum values. So, I'm confident with this solution.

💡Okay, so I have this geometry problem here. It says that through the vertices of triangle ABC, which is inscribed in a circle, parallel lines are drawn, each of which intersects the circle again at points A1, B1, and C1 respectively. I need to prove that triangles ABC and A1B1C1 are congruent.Hmm, let me try to visualize this. Triangle ABC is inscribed in a circle, so it's a cyclic triangle. That means all its vertices lie on the circumference of the circle. Now, through each vertex, we're drawing a line parallel to something. Wait, the problem says "parallel lines are drawn," but it doesn't specify what they're parallel to. Oh, wait, it says "each of which intersects the circle again at points A1, B1, and C1 respectively." So, does that mean each line is drawn through a vertex and is parallel to the opposite side? Or is it that each line is drawn through a vertex and is parallel to some other line?Wait, maybe I misread. Let me check again. It says, "Through the vertices of triangle ABC, which is inscribed in a circle, parallel lines are drawn, each of which intersects the circle again at points A1, B1, and C1 respectively." So, it seems that through each vertex, a line is drawn, and these lines are parallel to each other? Or are they each parallel to something else?Wait, the wording is a bit unclear. It says "parallel lines are drawn," so maybe each line is parallel to the corresponding side of the triangle? Or perhaps each line is parallel to a specific direction. Hmm.Wait, maybe it's that through each vertex, a line is drawn parallel to the opposite side. That would make sense because then each line would be parallel to one side of the triangle. Let me assume that for now. So, through vertex A, we draw a line parallel to BC, which intersects the circle again at A1. Similarly, through vertex B, a line parallel to AC intersects the circle again at B1, and through vertex C, a line parallel to AB intersects the circle again at C1.Okay, so now we have triangle ABC and triangle A1B1C1, both inscribed in the same circle. I need to show that these two triangles are congruent.Alright, so first, since ABC is inscribed in the circle, all its vertices lie on the circumference. Similarly, A1, B1, and C1 are points where the parallel lines through A, B, and C intersect the circle again. So, A1 is the second intersection point of the line through A parallel to BC with the circle. Similarly for B1 and C1.Now, since the lines are parallel to the sides of the triangle, the arcs subtended by these lines should be equal because of the properties of parallel chords in a circle. Wait, in a circle, if two chords are parallel, then the arcs they subtend are equal. So, if AA1 is parallel to BC, then the arc from A to A1 should be equal to the arc from B to C.Wait, is that correct? Let me think. If two chords are parallel, the arcs they subtend are equal. So, if AA1 is parallel to BC, then the arc AA1 should be equal to the arc BC. Similarly, the arc BB1 should be equal to the arc AC, and the arc CC1 should be equal to the arc AB.Wait, but in a circle, the length of an arc is proportional to the central angle subtended by it. So, if two chords are parallel, their subtended arcs are equal. Therefore, the central angles corresponding to those arcs are equal.So, if AA1 is parallel to BC, then the central angle for arc AA1 is equal to the central angle for arc BC. Similarly, the central angle for arc BB1 is equal to the central angle for arc AC, and the central angle for arc CC1 is equal to the central angle for arc AB.But in triangle ABC, the central angles correspond to the sides. Wait, no, in a triangle inscribed in a circle, the central angles correspond to twice the inscribed angles. So, for example, the central angle for arc BC is twice the angle at A.Wait, maybe I need to think in terms of inscribed angles. The inscribed angle subtended by an arc is half the measure of the central angle subtended by the same arc.So, if AA1 is parallel to BC, then the arc AA1 is equal to the arc BC. Therefore, the central angle for arc AA1 is equal to the central angle for arc BC. Similarly, the central angle for arc BB1 is equal to the central angle for arc AC, and the central angle for arc CC1 is equal to the central angle for arc AB.Therefore, the central angles for arcs AA1, BB1, and CC1 are equal to the central angles for arcs BC, AC, and AB respectively.But in triangle ABC, the central angles for arcs BC, AC, and AB correspond to twice the angles at A, B, and C respectively. So, the central angle for arc BC is 2A, for arc AC is 2B, and for arc AB is 2C.Therefore, the central angles for arcs AA1, BB1, and CC1 are 2A, 2B, and 2C respectively.Wait, but in triangle ABC, the sum of angles A, B, and C is 180 degrees, so the sum of the central angles would be 360 degrees, which makes sense because they complete the circle.Now, considering triangle A1B1C1, which is formed by points A1, B1, and C1. Since these points are obtained by drawing lines through A, B, and C parallel to the opposite sides, and intersecting the circle again, we can analyze the central angles for arcs A1B1, B1C1, and C1A1.Wait, but actually, the central angles for arcs A1B1, B1C1, and C1A1 would correspond to the central angles for arcs BC, AC, and AB respectively, because of the parallelism.Wait, no, let me think again. The central angle for arc A1B1 would correspond to the angle at the center between points A1 and B1. Since A1 is obtained by drawing a line through A parallel to BC, and B1 is obtained by drawing a line through B parallel to AC, the arc A1B1 would correspond to the central angle equal to the central angle for arc BC plus the central angle for arc AC?Wait, maybe not. Let me try a different approach.Since AA1 is parallel to BC, the arc AA1 is equal to arc BC. Similarly, BB1 is parallel to AC, so arc BB1 is equal to arc AC, and CC1 is parallel to AB, so arc CC1 is equal to arc AB.Therefore, the arcs AA1, BB1, and CC1 are equal to arcs BC, AC, and AB respectively.Now, considering triangle A1B1C1, the arcs between A1, B1, and C1 would be the remaining arcs on the circle after subtracting arcs AA1, BB1, and CC1.Wait, but the circle is 360 degrees, so the arcs between A1, B1, and C1 would be the arcs that are not covered by AA1, BB1, and CC1.But since arcs AA1, BB1, and CC1 are equal to arcs BC, AC, and AB respectively, the arcs between A1, B1, and C1 would be equal to arcs AB, BC, and AC.Wait, that seems a bit confusing. Maybe I need to think about the positions of A1, B1, and C1.Alternatively, perhaps considering the triangle A1B1C1, since each of its sides corresponds to a chord that is parallel to a side of triangle ABC, the arcs subtended by these sides would be equal to the arcs subtended by the sides of ABC.Therefore, the central angles for the sides of A1B1C1 would be equal to the central angles for the sides of ABC, meaning that the triangles are congruent.Wait, but I need to make this more precise.Alternatively, maybe using the fact that the triangles are similar because their corresponding angles are equal, and since they are inscribed in the same circle, they must be congruent.Wait, let's think about the angles of triangle A1B1C1.Since AA1 is parallel to BC, the angle at A1 in triangle A1B1C1 would be equal to the angle at B in triangle ABC, because of the parallel lines.Wait, no, that might not be correct. Let me think.In triangle ABC, angle at A is subtended by arc BC. In triangle A1B1C1, angle at A1 is subtended by arc B1C1. But since arc B1C1 is equal to arc BC, because BB1 is parallel to AC and CC1 is parallel to AB, then the angle at A1 would be equal to the angle at A in triangle ABC.Wait, is that correct?Wait, in a circle, the measure of an inscribed angle is half the measure of its subtended arc. So, if arc B1C1 is equal to arc BC, then the inscribed angle at A1 subtended by arc B1C1 is equal to the inscribed angle at A subtended by arc BC. Therefore, angle A1 is equal to angle A.Similarly, angle B1 is equal to angle B, and angle C1 is equal to angle C.Therefore, triangles ABC and A1B1C1 have equal corresponding angles, which means they are similar.But since they are inscribed in the same circle, their sides are chords of the same circle, and if their corresponding angles are equal, then their corresponding sides must be equal in length as well, because the length of a chord is determined by the central angle it subtends.Therefore, triangles ABC and A1B1C1 are congruent.Wait, that seems to make sense. So, to summarize:1. Through each vertex of triangle ABC, draw a line parallel to the opposite side, intersecting the circle again at A1, B1, and C1.2. Since these lines are parallel to the sides, the arcs they subtend are equal to the arcs subtended by the sides of ABC.3. Therefore, the inscribed angles of triangle A1B1C1 are equal to the inscribed angles of triangle ABC.4. Since the triangles are similar and inscribed in the same circle, their corresponding sides are equal, making them congruent.I think that's the gist of it. Maybe I should write it out more formally, but that's the basic idea.

💡Okay, so I have this math problem here where the operation "Y" is redefined. It says that for any two numbers a and b, a Y b is equal to (2a² - 3ab + b²) squared. And I need to find the value of 4 Y 3. Hmm, let's break this down step by step.First, I need to understand what the operation "Y" does. It's not the usual addition or multiplication; it's a custom operation defined by this specific formula. So, whenever I see "a Y b," I should replace it with (2a² - 3ab + b²) squared. That means I have to compute the expression inside the parentheses first and then square the result.Alright, so for 4 Y 3, I substitute a with 4 and b with 3 in the formula. Let me write that out:4 Y 3 = (2*(4)² - 3*(4)*(3) + (3)²)².Now, let's compute each part step by step. I'll start with the terms inside the parentheses.First term: 2*(4)². Okay, 4 squared is 16, and multiplying that by 2 gives me 32.Second term: -3*(4)*(3). Let's compute that. 4 times 3 is 12, and then multiplying by 3 gives 36. But since there's a negative sign in front, it becomes -36.Third term: (3)². That's straightforward; 3 squared is 9.So now, putting it all together inside the parentheses:32 (from the first term) minus 36 (from the second term) plus 9 (from the third term).Let me do the subtraction first: 32 - 36. That gives me -4. Then, adding 9 to that: -4 + 9 equals 5.So, the expression inside the parentheses simplifies to 5. Now, I need to square this result. Squaring 5 gives me 25.Wait, let me double-check my calculations to make sure I didn't make any mistakes. First term: 2*(4)² = 2*16 = 32. That seems correct.Second term: -3*(4)*(3) = -3*12 = -36. Yep, that's right.Third term: (3)² = 9. Correct.Adding them up: 32 - 36 + 9. 32 - 36 is -4, and -4 + 9 is indeed 5. Squaring 5 gives 25. Okay, that seems solid.I was a bit worried about the signs, especially with the negative in front of the second term. But breaking it down step by step helped me avoid any mistakes there. It's important to handle each operation carefully, especially when dealing with negative numbers.Another thing I noticed is that the expression inside the parentheses is a quadratic in terms of a and b. Maybe there's a way to factor it or simplify it before plugging in the numbers? Let me see.Looking at 2a² - 3ab + b², I wonder if this can be factored. Let's try factoring it.We have 2a² - 3ab + b². Let's see if this factors into two binomials.The general form would be (ma + nb)(pa + qb). Multiplying these out gives mpa² + (mq + np)ab + nqb².Comparing coefficients:mp = 2 (coefficient of a²)mq + np = -3 (coefficient of ab)nq = 1 (coefficient of b²)Since nq = 1, the possible values for n and q are both 1 or both -1. Let's try n=1 and q=1.Then, we have mp = 2 and mq + np = -3.So, m and p are factors of 2. Possible pairs are (1,2) or (2,1). Let's try m=2 and p=1.Then, mq + np = 2*1 + 1*1 = 2 + 1 = 3, which is not equal to -3. Hmm, that's not right.What if m=1 and p=2?Then, mq + np = 1*1 + 2*1 = 1 + 2 = 3, still not -3.Maybe n and q are both -1? Let's try that.So, n=-1 and q=-1.Then, mp = 2, and mq + np = m*(-1) + p*(-1) = -m - p = -3.So, -m - p = -3 implies m + p = 3.We also have mp = 2. So, we need two numbers m and p such that m + p = 3 and mp = 2.The factors of 2 are 1 and 2. 1 + 2 = 3, which fits.So, m=1 and p=2.Therefore, the factors are (1a + (-1)b)(2a + (-1)b) = (a - b)(2a - b).Let me check that:(a - b)(2a - b) = a*(2a - b) - b*(2a - b) = 2a² - ab - 2ab + b² = 2a² - 3ab + b².Yes! So, 2a² - 3ab + b² factors into (a - b)(2a - b). That might be useful for future problems, but for this specific case, since we're plugging in numbers, it might not save us much time.But it's good to know that the expression can be factored, which might help in simplifying or understanding the operation better.So, going back to our original problem, 4 Y 3 = (2*(4)² - 3*(4)*(3) + (3)²)² = (32 - 36 + 9)² = (5)² = 25.Just to make sure, let me compute it again without factoring.Compute each term:2*(4)^2 = 2*16 = 32-3*(4)*(3) = -3*12 = -36(3)^2 = 9Adding them together: 32 - 36 + 9 = (32 + 9) - 36 = 41 - 36 = 5Square of 5 is 25.Yep, same result. So, whether I factor it or compute directly, I end up with 25.I think I'm confident that 4 Y 3 equals 25. It's always good to verify the calculations, especially when dealing with custom operations that aren't as straightforward as regular arithmetic.Another thing I could consider is plugging in different values for a and b to see how this operation behaves. For example, if a = b, what happens?Let's say a = b = x. Then, x Y x = (2x² - 3x*x + x²)^2 = (2x² - 3x² + x²)^2 = (0)^2 = 0.Interesting, so when a and b are equal, the operation yields zero. That's a unique property of this operation.What if a = 0? Then, 0 Y b = (2*0² - 3*0*b + b²)^2 = (0 - 0 + b²)^2 = (b²)^2 = b⁴.Similarly, if b = 0, then a Y 0 = (2a² - 0 + 0)^2 = (2a²)^2 = 4a⁴.So, the operation has different behaviors depending on the values of a and b. It's not symmetric in the sense that swapping a and b might not give the same result, unless the expression inside is symmetric.Wait, let's check if the expression inside is symmetric. The expression is 2a² - 3ab + b². If we swap a and b, we get 2b² - 3ba + a², which is not the same as the original expression because 2a² vs a² and 2b² vs b². So, it's not symmetric. Therefore, a Y b is not necessarily equal to b Y a.For example, let's compute 3 Y 4 and see if it's the same as 4 Y 3.Compute 3 Y 4:(2*(3)^2 - 3*(3)*(4) + (4)^2)^2 = (2*9 - 36 + 16)^2 = (18 - 36 + 16)^2 = (-2)^2 = 4.Wait, that's different from 4 Y 3, which was 25. So, indeed, a Y b is not equal to b Y a. That's an important observation.So, in our original problem, 4 Y 3 is 25, but 3 Y 4 is 4. That shows the operation is not commutative. It's essential to pay attention to the order of a and b when performing this operation.Another aspect to consider is whether this operation has any real-world applications or if it's purely theoretical. It might be used in certain algebraic structures or specific mathematical contexts where such operations are defined for particular purposes.But for now, focusing on the problem at hand, I think I've thoroughly checked and verified that 4 Y 3 equals 25. I don't see any mistakes in my calculations, and the step-by-step approach helped me ensure accuracy.In summary, by substituting a = 4 and b = 3 into the given operation's formula, carefully computing each term, and verifying the result through multiple methods, I can confidently conclude that 4 Y 3 is 25.

💡Okay, so I have this problem here about finding the vertex of a parabola given by the equation ( y = frac{1}{3}(x - 7)^2 + 5 ). Hmm, I remember that the vertex form of a parabola is usually written as ( y = a(x - h)^2 + k ), where ( (h, k) ) is the vertex. So, in this case, I need to identify ( h ) and ( k ) from the given equation.Let me look at the equation again: ( y = frac{1}{3}(x - 7)^2 + 5 ). Comparing this to the standard vertex form ( y = a(x - h)^2 + k ), I can see that ( a = frac{1}{3} ), ( h = 7 ), and ( k = 5 ). So, the vertex should be at ( (h, k) ), which would be ( (7, 5) ).Wait, but let me double-check to make sure I'm not making a mistake. Sometimes, I mix up the signs. The equation is ( (x - 7) ), so that means the vertex is shifted 7 units to the right along the x-axis, right? And then it's shifted up by 5 units because of the ( +5 ) at the end. So, that would definitely place the vertex at ( (7, 5) ).Let me also recall that the vertex form is handy because it directly gives you the vertex without having to complete the square or use calculus. Since the equation is already in vertex form, I don't need to do any additional steps. I just need to read off the values of ( h ) and ( k ).Just to visualize, if I were to graph this parabola, it would open upwards because the coefficient ( a = frac{1}{3} ) is positive. The vertex is the lowest point on the parabola in this case. So, the point ( (7, 5) ) is the minimum point of the parabola.I think I'm confident now that the vertex is at ( (7, 5) ). Let me check the answer choices:A: ( (7, -5) ) B: ( (-7, -5) ) C: ( (7, 5) ) D: ( (-7, 5) )So, option C matches my conclusion. I don't think I made any mistakes here, but just to be thorough, let's consider if there's any possibility I misread the equation. The equation is ( y = frac{1}{3}(x - 7)^2 + 5 ), so no, I didn't misread the signs or the numbers. It's definitely ( x - 7 ) and ( +5 ).Another way to think about it is to consider the transformations applied to the parent function ( y = x^2 ). The parent function has its vertex at the origin ( (0, 0) ). Here, we have a horizontal shift of 7 units to the right and a vertical shift of 5 units up. So, starting from ( (0, 0) ), moving 7 right and 5 up lands us at ( (7, 5) ).I think that's solid. I don't see any reason to doubt this answer. It all lines up with what I know about vertex form and transformations of functions.