💡First, I need to find the missing fraction that, when added to the sum of the given fractions, equals 0.13333333333333333.I'll start by converting all the given fractions to decimals to simplify the addition process.Next, I'll add these decimal values together to find the current sum of the known fractions.After obtaining the sum of the known fractions, I'll subtract this sum from the total sum (0.13333333333333333) to determine the value of the missing fraction.Finally, I'll convert the resulting decimal value of the missing fraction back to a fraction to present the answer in the required format.

💡Alright, so I have this problem with seven variables and three equations. The equations involve these variables multiplied by squares of integers, and I need to find the value of another similar expression. Hmm, okay, let me try to make sense of this.First, let me write down the given equations to see if I can spot a pattern or figure out a way to approach this.The equations are:1. ( x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 = 5 )2. ( 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 = 20 )3. ( 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 = 145 )And I need to find:( 16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7 )Looking at the coefficients, they seem to be perfect squares. The first equation has coefficients (1^2, 2^2, 3^2, ldots, 7^2). The second equation has coefficients (2^2, 3^2, 4^2, ldots, 8^2), and the third has (3^2, 4^2, 5^2, ldots, 9^2). So each subsequent equation shifts the squares by one.That makes me think that maybe there's a function or a sequence here that relates to these squares. Maybe if I model this as a function of some variable, I can find a pattern or a formula.Let me define a function ( f(k) ) such that:( f(k) = (k+1)^2x_1 + (k+2)^2x_2 + (k+3)^2x_3 + ldots + (k+7)^2x_7 )So, for ( k = 0 ), ( f(0) = 1^2x_1 + 2^2x_2 + 3^2x_3 + ldots + 7^2x_7 = 5 )For ( k = 1 ), ( f(1) = 2^2x_1 + 3^2x_2 + 4^2x_3 + ldots + 8^2x_7 = 20 )For ( k = 2 ), ( f(2) = 3^2x_1 + 4^2x_2 + 5^2x_3 + ldots + 9^2x_7 = 145 )And I need to find ( f(3) = 4^2x_1 + 5^2x_2 + 6^2x_3 + ldots + 10^2x_7 )So, if I can model ( f(k) ) as a quadratic function, maybe I can find its coefficients and then compute ( f(3) ).Let me assume that ( f(k) ) is a quadratic function in terms of ( k ). So, let's say:( f(k) = Ak^2 + Bk + C )Given that, I can plug in the known values of ( f(0) ), ( f(1) ), and ( f(2) ) to form a system of equations.From ( f(0) = 5 ):( A(0)^2 + B(0) + C = 5 )( C = 5 )From ( f(1) = 20 ):( A(1)^2 + B(1) + C = 20 )( A + B + 5 = 20 )( A + B = 15 ) --- Equation 1From ( f(2) = 145 ):( A(2)^2 + B(2) + C = 145 )( 4A + 2B + 5 = 145 )( 4A + 2B = 140 )Divide both sides by 2:( 2A + B = 70 ) --- Equation 2Now, I have two equations:1. ( A + B = 15 )2. ( 2A + B = 70 )Let me subtract Equation 1 from Equation 2 to eliminate ( B ):( (2A + B) - (A + B) = 70 - 15 )( A = 55 )Now, substitute ( A = 55 ) back into Equation 1:( 55 + B = 15 )( B = 15 - 55 )( B = -40 )So, now I have ( A = 55 ), ( B = -40 ), and ( C = 5 ).Therefore, the function ( f(k) ) is:( f(k) = 55k^2 - 40k + 5 )Now, to find ( f(3) ):( f(3) = 55(3)^2 - 40(3) + 5 )( f(3) = 55(9) - 120 + 5 )( f(3) = 495 - 120 + 5 )( f(3) = 495 - 120 = 375 )( 375 + 5 = 380 )So, ( f(3) = 380 ).Wait, let me double-check my calculations to make sure I didn't make a mistake.First, ( 55 times 9 ). 55 times 9 is 495, that's correct.Then, ( -40 times 3 = -120 ), that's correct.Adding 5: 495 - 120 is 375, plus 5 is 380. Yep, that seems right.So, I think the value of ( 16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7 ) is 380.I wonder if there's another way to approach this problem, maybe by solving the system of equations directly. But since we have seven variables and only three equations, it's underdetermined. However, by recognizing the pattern and modeling it as a quadratic function, we can bypass solving for each variable individually, which is smart.I think this method works because the way the coefficients shift by one each time suggests a polynomial relationship. Since we have three points, we can fit a quadratic curve through them, which gives us the coefficients to predict the next value.Yeah, I feel confident about this answer. It makes sense, and the calculations check out.**Final Answer**boxed{380}

💡Okay, so I have this problem where I need to find g(f(3)). The functions given are f(x) = x³ - 2x + 1 and g(x) = 3x² + x + 2. Hmm, let me break this down step by step.First, I remember that to find g(f(3)), I need to compute f(3) first and then plug that result into the function g. So, let me start by finding f(3).The function f(x) is x³ - 2x + 1. So, if I substitute x with 3, it should be 3³ - 2*3 + 1. Let me calculate that:3³ is 27, right? Then, 2*3 is 6. So, putting it all together: 27 - 6 + 1. That simplifies to 27 - 6 is 21, and then 21 + 1 is 22. So, f(3) equals 22. Got that down.Now, I need to find g(f(3)), which is g(22). So, I have to plug 22 into the function g(x). The function g(x) is 3x² + x + 2. Let me substitute x with 22:First, calculate 22 squared. 22 times 22 is... let me think, 20 times 20 is 400, and then 2 times 20 is 40, and 2 times 20 again is another 40, and 2 times 2 is 4. Wait, no, that's not the right way. Maybe I should just multiply 22 by 22 directly.22 times 20 is 440, and 22 times 2 is 44, so adding those together, 440 + 44 is 484. So, 22 squared is 484.Now, multiply that by 3. 3 times 484. Let me compute that step by step. 3 times 400 is 1200, 3 times 80 is 240, and 3 times 4 is 12. Adding those together: 1200 + 240 is 1440, plus 12 is 1452. So, 3 times 484 is 1452.Next, I need to add the other terms in g(x). The function is 3x² + x + 2, so after calculating 3x², which is 1452, I need to add x, which is 22, and then add 2.So, 1452 + 22 is... let me see, 1452 + 20 is 1472, and then +2 is 1474. Then, adding the last 2, that would be 1474 + 2, which is 1476.Wait, hold on, did I do that right? Let me double-check. 3x² is 1452, then adding x which is 22 gives 1452 + 22 = 1474, and then adding 2 gives 1476. Yeah, that seems correct.So, putting it all together, f(3) is 22, and then g(22) is 1476. Therefore, g(f(3)) is 1476.I think that's all. I don't see any mistakes in my calculations, but just to be thorough, let me go through each step again.First, f(3): 3³ is 27, minus 2*3 is 6, plus 1. So, 27 - 6 is 21, plus 1 is 22. Correct.Then, g(22): 3*(22)² + 22 + 2. 22 squared is 484, multiplied by 3 is 1452. Then, adding 22 gives 1474, and adding 2 gives 1476. Yep, that all checks out.I don't think I made any errors here. So, the final answer should be 1476.

💡First, I need to determine the distance to the vacation destination, which I'll denote as ( D ) miles.Abel is traveling at a speed of 50 mph, so the time it takes him to reach the destination is ( T_{Abel} = frac{D}{50} ) hours.Alice starts one hour later than Abel and travels at a speed of 40 mph. Therefore, the time it takes Alice to reach the destination is ( T_{Alice} = frac{D}{40} ) hours.According to the problem, Abel reaches the destination 360 minutes earlier than Alice. Since 360 minutes is equivalent to 6 hours, I can set up the equation:[ T_{Alice} = T_{Abel} + 6 ]Substituting the expressions for ( T_{Abel} ) and ( T_{Alice} ) into the equation:[ frac{D}{40} = frac{D}{50} + 6 ]To solve for ( D ), I'll find a common denominator for the fractions, which is 200. Rewriting the equation:[ frac{5D}{200} = frac{4D}{200} + 6 ]Multiplying through by 200 to eliminate the denominators:[ 5D = 4D + 1200 ]Subtracting ( 4D ) from both sides:[ D = 1200 ]Therefore, the vacation destination is 1200 miles away.

💡Okay, so I have this problem about two sequences, {a_n} and {b_n}. Let me try to figure it out step by step.First, part (I) says that the sequence {a_n} satisfies a_1 = 1 and a_{n+1} + 2a_n = 0. I need to find the general term formula and the sum of the first n terms, S_n.Hmm, okay, so starting with a_1 = 1. The recurrence relation is a_{n+1} + 2a_n = 0, which I can rewrite as a_{n+1} = -2a_n. That looks like a geometric sequence because each term is a constant multiple of the previous term. The common ratio seems to be -2.So, for a geometric sequence, the general term is a_n = a_1 * r^{n-1}, where r is the common ratio. Plugging in the values, a_n = 1 * (-2)^{n-1}, so a_n = (-2)^{n-1}. That should be the general term.Now, for the sum of the first n terms, S_n. The formula for the sum of a geometric series is S_n = a_1 * (1 - r^n) / (1 - r). Here, a_1 is 1 and r is -2. So plugging in, S_n = (1 - (-2)^n) / (1 - (-2)) = (1 - (-2)^n) / 3. That should be the sum.Alright, moving on to part (II). It says that the arithmetic sequence {b_n} satisfies b_1 = a_4 and b_2 = a_2 - a_3. I need to find b_{37} and determine which term in {a_n} is equal to b_{37}.First, let's find a_4, a_2, and a_3 using the general term we found earlier.a_4 = (-2)^{4-1} = (-2)^3 = -8.a_2 = (-2)^{2-1} = (-2)^1 = -2.a_3 = (-2)^{3-1} = (-2)^2 = 4.So, b_1 = a_4 = -8.b_2 = a_2 - a_3 = (-2) - 4 = -6.Now, since {b_n} is an arithmetic sequence, the difference between consecutive terms is constant. Let's find the common difference, d.d = b_2 - b_1 = (-6) - (-8) = 2.So, the common difference is 2. The general term for an arithmetic sequence is b_n = b_1 + (n - 1)d.Plugging in the values, b_n = -8 + (n - 1)*2 = -8 + 2n - 2 = 2n - 10.So, the general term is b_n = 2n - 10.Now, to find b_{37}, plug in n = 37:b_{37} = 2*37 - 10 = 74 - 10 = 64.Now, I need to determine which term in {a_n} is equal to b_{37}, which is 64.So, set a_n = 64 and solve for n.a_n = (-2)^{n-1} = 64.Hmm, 64 is a positive number, and (-2)^{n-1} can be positive or negative depending on the exponent. Since 64 is positive, the exponent must be even because (-2)^{even} is positive.So, let's solve (-2)^{n-1} = 64.First, note that 64 is 2^6. So, (-2)^{n-1} = 2^6.But (-2)^{n-1} is equal to (-1)^{n-1} * 2^{n-1}.So, we have (-1)^{n-1} * 2^{n-1} = 2^6.Since 2^{n-1} must be positive, and 2^6 is positive, (-1)^{n-1} must be positive as well. Therefore, n-1 must be even, so n is odd.Now, equate the magnitudes: 2^{n-1} = 2^6. Therefore, n - 1 = 6, so n = 7.Therefore, a_7 = 64, which is equal to b_{37}.Let me double-check:a_7 = (-2)^{7-1} = (-2)^6 = 64. Yep, that's correct.And b_{37} = 2*37 - 10 = 74 - 10 = 64. That's correct too.So, everything seems to check out.

💡Okay, so I have this problem here. Let me try to understand it step by step. First, the problem says: Let O be the origin of coordinates, C be the center of the circle defined by the equation ((x - 2)^2 + y^2 = 3). There's a point M((x, y)) on this circle that satisfies the condition (overrightarrow{OM} cdot overrightarrow{CM} = 0). We need to find the value of (frac{y}{x}) and choose the correct option from the given choices.Alright, so let me break this down. 1. **Understanding the Circle:** The equation of the circle is ((x - 2)^2 + y^2 = 3). From this, I can tell that the center of the circle, point C, is at (2, 0) because the standard form of a circle is ((x - h)^2 + (y - k)^2 = r^2), where (h, k) is the center and r is the radius. So here, h = 2, k = 0, and the radius squared is 3, which means the radius is (sqrt{3}).2. **Understanding the Vectors:** The vectors in question are (overrightarrow{OM}) and (overrightarrow{CM}). - (overrightarrow{OM}) is the vector from the origin O(0, 0) to the point M(x, y). So, this vector can be represented as ((x, y)). - (overrightarrow{CM}) is the vector from the center C(2, 0) to the point M(x, y). So, this vector can be represented as ((x - 2, y - 0)) or simply ((x - 2, y)).3. **Dot Product Condition:** The condition given is that the dot product of (overrightarrow{OM}) and (overrightarrow{CM}) is zero. In vector terms, this means: [ overrightarrow{OM} cdot overrightarrow{CM} = 0 ] Substituting the vectors we have: [ (x, y) cdot (x - 2, y) = 0 ] Calculating the dot product: [ x(x - 2) + y cdot y = 0 ] Simplifying: [ x^2 - 2x + y^2 = 0 ] So, this is the equation we get from the dot product condition.4. **Using the Circle Equation:** We also know that point M lies on the circle, so it must satisfy the circle's equation: [ (x - 2)^2 + y^2 = 3 ] Let me expand this equation: [ x^2 - 4x + 4 + y^2 = 3 ] Simplifying: [ x^2 - 4x + y^2 + 4 - 3 = 0 ] Which becomes: [ x^2 - 4x + y^2 + 1 = 0 ] So now, I have two equations: - From the dot product: (x^2 - 2x + y^2 = 0) - From the circle: (x^2 - 4x + y^2 + 1 = 0)5. **Subtracting the Equations:** Let me subtract the first equation from the second to eliminate (x^2) and (y^2): [ (x^2 - 4x + y^2 + 1) - (x^2 - 2x + y^2) = 0 - 0 ] Simplifying: [ -4x + 1 - (-2x) = 0 ] Which is: [ -4x + 1 + 2x = 0 ] Combining like terms: [ -2x + 1 = 0 ] Solving for x: [ -2x = -1 implies x = frac{1}{2} ] So, x is 1/2.6. **Finding y:** Now that I have x = 1/2, I can substitute this back into one of the equations to find y. Let me use the dot product equation: [ x^2 - 2x + y^2 = 0 ] Plugging in x = 1/2: [ left(frac{1}{2}right)^2 - 2 cdot frac{1}{2} + y^2 = 0 ] Calculating each term: - (left(frac{1}{2}right)^2 = frac{1}{4}) - (2 cdot frac{1}{2} = 1) So, substituting: [ frac{1}{4} - 1 + y^2 = 0 ] Simplifying: [ -frac{3}{4} + y^2 = 0 implies y^2 = frac{3}{4} ] Taking square roots: [ y = pm frac{sqrt{3}}{2} ] So, y can be either (frac{sqrt{3}}{2}) or (-frac{sqrt{3}}{2}).7. **Calculating (frac{y}{x}):** Now, we need to find the ratio (frac{y}{x}). Since x is 1/2 and y is (pm frac{sqrt{3}}{2}), let's compute this: [ frac{y}{x} = frac{pm frac{sqrt{3}}{2}}{frac{1}{2}} = pm sqrt{3} ] So, the ratio (frac{y}{x}) is either (sqrt{3}) or (-sqrt{3}).8. **Looking at the Options:** The options given are: - A: (frac{sqrt{3}}{3}) - B: (frac{sqrt{3}}{3}) or (-frac{sqrt{3}}{3}) - C: (sqrt{3}) - D: (sqrt{3}) or (-sqrt{3}) From my calculation, the ratio is (sqrt{3}) or (-sqrt{3}), which corresponds to option D.9. **Double-Checking:** Let me just verify my steps to make sure I didn't make a mistake. - I correctly identified the center C as (2, 0) and the radius as (sqrt{3}). - I correctly wrote the vectors (overrightarrow{OM}) and (overrightarrow{CM}). - The dot product calculation led me to (x^2 - 2x + y^2 = 0), which seems right. - The circle equation substitution also seems correct, leading to (x^2 - 4x + y^2 + 1 = 0). - Subtracting the two equations gave me x = 1/2, which seems correct. - Plugging x = 1/2 back into the dot product equation gave me y = (pm frac{sqrt{3}}{2}), which seems correct. - Therefore, the ratio (frac{y}{x}) is indeed (pm sqrt{3}). So, I think my solution is correct.10. **Alternative Approach:** Wait, the initial thought process mentioned something about the tangent line. Maybe I can think about it that way too. If (overrightarrow{OM} cdot overrightarrow{CM} = 0), then vectors OM and CM are perpendicular. So, OM is perpendicular to CM. Since C is the center of the circle, and M is a point on the circle, the line CM is the radius. So, if OM is perpendicular to CM, then OM is tangent to the circle at point M. So, OM is a tangent line to the circle. That's another way to think about it. If OM is a tangent to the circle, then the distance from the center C to the line OM must be equal to the radius of the circle. Let me try this approach. Let the equation of line OM be y = kx, since it passes through the origin. The distance from point C(2, 0) to the line y = kx is given by: [ frac{|k cdot 2 - 0|}{sqrt{k^2 + 1}}} = frac{|2k|}{sqrt{k^2 + 1}} ] Since this distance must equal the radius (sqrt{3}), we have: [ frac{|2k|}{sqrt{k^2 + 1}} = sqrt{3} ] Squaring both sides to eliminate the square roots: [ frac{4k^2}{k^2 + 1} = 3 ] Multiply both sides by (k^2 + 1): [ 4k^2 = 3(k^2 + 1) ] Expanding: [ 4k^2 = 3k^2 + 3 ] Subtracting (3k^2) from both sides: [ k^2 = 3 ] Taking square roots: [ k = pm sqrt{3} ] Therefore, the slope of line OM is (pm sqrt{3}), which is (frac{y}{x}), so (frac{y}{x} = pm sqrt{3}). This confirms the earlier result. So, regardless of the approach, the answer is D.11. **Conclusion:** Both methods lead to the same conclusion. The ratio (frac{y}{x}) is either (sqrt{3}) or (-sqrt{3}), so the correct answer is D.**Final Answer**boxed{D}