💡Okay, so I have this problem here about an arithmetic sequence. Let me try to understand it step by step. The sequence is called {a_n}, and it's an arithmetic sequence. The first term is 1, and the common difference is 2. I need to find the limit as n approaches infinity of S_n divided by a_n squared. Hmm, okay.First, let me recall what an arithmetic sequence is. An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference to the preceding term. In this case, the first term is 1, and the common difference is 2. So, the sequence would go 1, 3, 5, 7, 9, and so on. Each time, we add 2 to get the next term.Now, S_n is the sum of the first n terms of this sequence. So, I need to find a formula for S_n. I remember that the sum of the first n terms of an arithmetic sequence can be calculated using the formula:S_n = n/2 * (2a_1 + (n - 1)d)Where:- S_n is the sum of the first n terms,- a_1 is the first term,- d is the common difference,- n is the number of terms.Plugging in the values we have:a_1 = 1,d = 2.So, substituting these into the formula:S_n = n/2 * (2*1 + (n - 1)*2)Simplify inside the parentheses first:2*1 = 2,(n - 1)*2 = 2n - 2.So, adding those together:2 + 2n - 2 = 2n.Therefore, S_n = n/2 * 2n = n * n = n^2.Wait, that seems too straightforward. Let me double-check. The formula is S_n = n/2*(first term + last term). The last term, a_n, can be found using the formula for the nth term of an arithmetic sequence:a_n = a_1 + (n - 1)d.So, plugging in:a_n = 1 + (n - 1)*2 = 1 + 2n - 2 = 2n - 1.So, the last term is 2n - 1. Then, the sum S_n is n/2*(1 + (2n - 1)) = n/2*(2n) = n^2. Yep, that checks out. So, S_n is indeed n squared.Now, the problem asks for the limit as n approaches infinity of S_n divided by a_n squared. So, let's write that out:Limit = lim (n→∞) [S_n / (a_n)^2]We already have S_n = n^2 and a_n = 2n - 1. So, substituting these in:Limit = lim (n→∞) [n^2 / (2n - 1)^2]Hmm, okay. So, I need to compute this limit. Let me think about how to approach this. When dealing with limits at infinity, especially involving polynomials, it's often helpful to look at the highest degree terms in the numerator and denominator because they dominate the behavior as n becomes very large.In the numerator, we have n^2, which is a quadratic term. In the denominator, we have (2n - 1)^2. Let me expand that to see what it looks like:(2n - 1)^2 = (2n)^2 - 2*2n*1 + 1^2 = 4n^2 - 4n + 1.So, the denominator becomes 4n^2 - 4n + 1. Now, the numerator is n^2, and the denominator is 4n^2 - 4n + 1.So, the expression becomes:Limit = lim (n→∞) [n^2 / (4n^2 - 4n + 1)]To evaluate this limit, I can divide both the numerator and the denominator by n^2, the highest power of n in the denominator. Let's do that:Limit = lim (n→∞) [ (n^2 / n^2) / ( (4n^2 - 4n + 1)/n^2 ) ]Simplify each term:Numerator: n^2 / n^2 = 1.Denominator: (4n^2 - 4n + 1)/n^2 = 4 - 4/n + 1/n^2.So, the expression simplifies to:Limit = lim (n→∞) [1 / (4 - 4/n + 1/n^2)]Now, let's evaluate the limit as n approaches infinity. As n becomes very large, the terms with 1/n and 1/n^2 will approach zero. So:Limit = 1 / (4 - 0 + 0) = 1/4.Therefore, the limit is 1/4.Wait, let me make sure I didn't make any mistakes here. I started by finding S_n, which is n^2, and a_n, which is 2n - 1. Then, I substituted them into the limit expression and expanded the denominator. Dividing numerator and denominator by n^2 was a good move because it simplifies the expression and makes it easier to see the behavior as n approaches infinity.Another way to think about this is by considering the leading coefficients. In the numerator, the leading term is n^2, and in the denominator, after expanding, the leading term is 4n^2. So, the ratio of the leading coefficients is 1/4, which is the limit. This is a quicker way to find the limit without expanding, but it's good that I did the expansion to verify.So, I think my answer is correct. The limit is 1/4.**Final Answer**The value of the limit is boxed{dfrac{1}{4}}.

💡Okay, so I have this problem where two cars, Car A and Car B, start at the same time from points A and B, respectively, heading towards each other. They meet for the first time at a point that's 60 kilometers away from A. Then, after they meet, they keep going to their destinations—Car A goes to B and Car B goes to A—and then they immediately turn around and start heading back. They meet again at a point that's 50 kilometers away from B. I need to find the distance between A and B.Alright, let me try to visualize this. So, first, both cars start moving towards each other. They meet somewhere in between, and that meeting point is 60 km from A. That means Car A has traveled 60 km when they meet. Since they started at the same time and met at the same time, the time taken for both cars to meet is the same.Let me denote the distance between A and B as S kilometers. So, when they meet the first time, Car A has gone 60 km, and Car B has gone S - 60 km. That makes sense because the total distance between A and B is S, so if Car A went 60 km, Car B must have covered the remaining S - 60 km.Now, after meeting, they continue to their respective destinations. So, Car A, which has 60 km to go to reach B, will cover that distance. Similarly, Car B, which has S - 60 km to go to reach A, will cover that distance. Then, they both turn around immediately and start heading back towards each other again.They meet again at a point 50 km from B. So, this second meeting point is closer to B than to A. I need to figure out how far S is based on this information.Let me think about the distances each car has traveled by the time they meet the second time. Starting from the first meeting point, Car A goes to B, which is S - 60 km away, and then turns around and comes back towards A. When they meet again, Car A has traveled an additional distance. Similarly, Car B goes to A, which is 60 km away, turns around, and comes back towards B, and meets Car A again.So, from the first meeting point to the second meeting point, Car A has traveled (S - 60) km to reach B and then some distance back towards A. Similarly, Car B has traveled 60 km to reach A and then some distance back towards B.Wait, but the second meeting point is 50 km from B. So, from B, Car A has traveled 50 km towards A when they meet the second time. That means, after reaching B, Car A turned around and went back 50 km. So, the total distance Car A has traveled from the first meeting point is (S - 60) km to B plus 50 km back towards A, which is (S - 60 + 50) km.Similarly, Car B, after reaching A, turned around and went back towards B. The second meeting point is 50 km from B, so from A, Car B has traveled (S - 50) km towards B. Wait, no. Let me think again.If the second meeting point is 50 km from B, then from A, it's S - 50 km. But Car B started from A after the first meeting, so the distance Car B traveled from A to the second meeting point is S - 50 km. But wait, no, because Car B went from the first meeting point to A, which is 60 km, and then turned around and went back towards B. So, the distance from A to the second meeting point is S - 50 km, which means Car B traveled (S - 50) km after turning around. But wait, that doesn't sound right.Let me clarify. The second meeting point is 50 km from B, so from A, it's S - 50 km. But Car B went from the first meeting point (which is 60 km from A) to A, which is 60 km, and then turned around and went back towards B. So, the distance from A to the second meeting point is S - 50 km, so Car B traveled (S - 50) km after turning around. Therefore, the total distance Car B traveled from the first meeting point is 60 km to A plus (S - 50) km back towards B, which is 60 + (S - 50) km.Similarly, Car A traveled from the first meeting point to B, which is S - 60 km, and then turned around and went back 50 km towards A, so the total distance Car A traveled from the first meeting point is (S - 60) + 50 km.Now, since both cars started moving at the same time and met at the first point, their speeds are constant, so the ratio of their speeds should be the same as the ratio of the distances they traveled before the first meeting. That is, the speed of Car A over the speed of Car B is equal to 60 over (S - 60).Similarly, after the first meeting, they continue to their destinations and then turn around. The time taken for them to meet again should be the same for both cars, so the ratio of the distances they traveled after the first meeting should also be the same as their speed ratio.So, the distance Car A traveled after the first meeting is (S - 60 + 50) km, and the distance Car B traveled after the first meeting is (60 + S - 50) km. Therefore, the ratio of these distances should be equal to the ratio of their speeds, which is 60/(S - 60).So, setting up the equation:60 / (S - 60) = (S - 60 + 50) / (60 + S - 50)Simplifying the numerator and denominator on the right side:(S - 60 + 50) = S - 10(60 + S - 50) = S + 10So, the equation becomes:60 / (S - 60) = (S - 10) / (S + 10)Now, cross-multiplying:60 * (S + 10) = (S - 60) * (S - 10)Expanding both sides:60S + 600 = S^2 - 10S - 60S + 600Simplifying the right side:S^2 - 70S + 600So, the equation is:60S + 600 = S^2 - 70S + 600Subtracting 60S + 600 from both sides:0 = S^2 - 130SFactoring:S(S - 130) = 0So, S = 0 or S = 130Since distance can't be zero, S = 130 km.Wait, let me double-check. If S is 130 km, then the first meeting point is 60 km from A, so 70 km from B. Then, after meeting, Car A goes 70 km to B and turns back, traveling 50 km towards A, so total from first meeting point: 70 + 50 = 120 km. Car B goes 60 km to A and turns back, traveling 80 km towards B (since 130 - 50 = 80 km from A). So, total from first meeting point: 60 + 80 = 140 km.Now, the ratio of their speeds should be 60/70 = 6/7, and the ratio of distances after first meeting is 120/140 = 12/14 = 6/7. So, it checks out.Therefore, the distance between A and B is 130 km.

💡First, I need to find the value of ( m ) such that the vector ( a + c ) is perpendicular to vector ( b ). To do this, I'll calculate the sum of vectors ( a ) and ( c ):[a + c = (1 + 2, 2m + m) = (3, 3m)]Next, since ( a + c ) is perpendicular to ( b ), their dot product must be zero:[(3, 3m) cdot (m + 1, 1) = 0][3(m + 1) + 3m times 1 = 0][3m + 3 + 3m = 0][6m + 3 = 0][6m = -3][m = -frac{1}{2}]Now that I have ( m = -frac{1}{2} ), I can find vector ( a ):[a = (1, 2m) = left(1, 2 times -frac{1}{2}right) = (1, -1)]Finally, I'll calculate the magnitude of vector ( a ):[|a| = sqrt{1^2 + (-1)^2} = sqrt{1 + 1} = sqrt{2}]

💡Okay, so I have this problem here: I need to find the phase shift and the vertical displacement of the graph of ( y = sin(5x - frac{pi}{2}) + 2 ). Hmm, I remember that when dealing with sine functions, there are some standard transformations that can be applied, like shifting the graph left or right, stretching or compressing it, and moving it up or down. First, let me recall the general form of a sine function with transformations. I think it's something like ( y = A sin(Bx + C) + D ). In this equation, ( A ) affects the amplitude, ( B ) affects the period, ( C ) is related to the phase shift, and ( D ) is the vertical displacement. So, in my problem, I need to identify ( C ) and ( D ) to find the phase shift and vertical displacement.Looking at my equation, ( y = sin(5x - frac{pi}{2}) + 2 ), I can rewrite it to match the general form. Let me see, ( A ) is 1 because there's no coefficient in front of the sine function. ( B ) is 5, which means the period is affected, but I don't need to worry about that right now. Then, inside the sine function, I have ( 5x - frac{pi}{2} ). That can be rewritten as ( 5(x - frac{pi}{10}) ) because if I factor out the 5, it becomes ( 5x - frac{pi}{2} = 5(x - frac{pi}{10}) ). So, the phase shift is the value that's being subtracted from ( x ) inside the sine function. In the general form, the phase shift is given by ( -frac{C}{B} ). Wait, let me make sure I have that right. If the equation is ( y = A sin(Bx - C) + D ), then the phase shift is ( frac{C}{B} ). Hmm, no, I think it's actually ( frac{C}{B} ) to the right. But in my case, the equation is ( y = sin(5x - frac{pi}{2}) + 2 ), which is like ( y = sin(Bx - C) + D ) with ( B = 5 ) and ( C = frac{pi}{2} ). So, the phase shift should be ( frac{C}{B} = frac{pi/2}{5} = frac{pi}{10} ). But wait, is it positive or negative? Since it's ( 5x - frac{pi}{2} ), it's a shift to the right by ( frac{pi}{10} ). So, the phase shift is ( frac{pi}{10} ) to the right.Now, for the vertical displacement. That's the value added outside the sine function, which in this case is ( +2 ). So, the graph is shifted upward by 2 units. That seems straightforward.Let me double-check my work. I rewrote the equation as ( sin(5(x - frac{pi}{10})) + 2 ), which shows that the graph is shifted to the right by ( frac{pi}{10} ) and up by 2. Yeah, that makes sense. The phase shift is ( frac{pi}{10} ) to the right, and the vertical displacement is 2 units up.Wait, but sometimes I get confused about whether the phase shift is positive or negative. If it's ( sin(Bx - C) ), is it a shift to the right or to the left? Let me think. If I have ( sin(B(x - phi)) ), then ( phi ) is the phase shift to the right. So, in my case, ( phi = frac{pi}{10} ), which is a shift to the right. So, the phase shift is positive ( frac{pi}{10} ).But in the original equation, it's ( 5x - frac{pi}{2} ), which is equivalent to ( 5(x - frac{pi}{10}) ). So, yes, it's a shift to the right by ( frac{pi}{10} ). Got it.And for the vertical displacement, it's just the constant term added outside the sine function, which is 2. So, the graph is moved up by 2 units. No confusion there.I think that's it. The phase shift is ( frac{pi}{10} ) to the right, and the vertical displacement is 2 units up.

💡First, I need to determine how many DVDs were sold based on the information provided.I'll start by defining variables for the quantities involved. Let ( C ) represent the number of CDs sold and ( D ) represent the number of DVDs sold.According to the problem, the store sold 1.6 times as many DVDs as CDs. This relationship can be expressed with the equation:[D = 1.6C]Additionally, the total number of DVDs and CDs sold that day was 273. This gives me another equation:[D + C = 273]Now, I can substitute the expression for ( D ) from the first equation into the second equation to solve for ( C ):[1.6C + C = 273]Combining like terms, I get:[2.6C = 273]To find the value of ( C ), I'll divide both sides of the equation by 2.6:[C = frac{273}{2.6} = 105]Now that I know the number of CDs sold (( C = 105 )), I can find the number of DVDs sold using the first equation:[D = 1.6 times 105 = 168]Therefore, the store sold 168 DVDs that day.

💡Okay, so I need to find the sine of 330 degrees. Hmm, I remember that sine is a trigonometric function that relates to the y-coordinate on the unit circle. Let me think about how to approach this.First, I know that 330 degrees is in the fourth quadrant because it's between 270 degrees and 360 degrees. In the fourth quadrant, sine values are negative because the y-coordinate is below the x-axis. That's something important to remember.Now, to find the sine of 330 degrees, I can use the concept of reference angles. A reference angle is the acute angle that the terminal side makes with the x-axis. For angles in the fourth quadrant, the reference angle is calculated by subtracting the given angle from 360 degrees. So, let me do that:360 degrees minus 330 degrees equals 30 degrees. So, the reference angle here is 30 degrees. I remember that 30 degrees is one of those special angles with known sine, cosine, and tangent values.I recall that sin(30 degrees) is 1/2. But wait, since 330 degrees is in the fourth quadrant where sine is negative, the sine of 330 degrees should be negative 1/2. Let me double-check that.Yes, in the unit circle, the coordinates for 30 degrees in the first quadrant are (√3/2, 1/2). So, for 330 degrees, which is 30 degrees below the x-axis in the fourth quadrant, the coordinates would be (√3/2, -1/2). Therefore, the sine, which is the y-coordinate, is -1/2.I think that's correct. Let me visualize the unit circle again to make sure I'm not making a mistake. Starting from 0 degrees, going counterclockwise, 330 degrees is just 30 degrees short of a full circle. So, it's in the fourth quadrant, and the y-coordinate is negative. The reference angle is 30 degrees, and sin(30 degrees) is 1/2, so with the negative sign, it's -1/2.I don't think I made any mistakes here. The logic seems solid. I used the reference angle correctly, accounted for the quadrant where the angle is located, and applied the appropriate sign to the sine value. So, I feel confident that sin(330 degrees) is indeed -1/2.