💡First, I recognize that in an arithmetic sequence, the difference between consecutive terms is constant. Given that ( a_3 = 1 ) and ( a_7 = 3 ), I can determine the common difference ( d ) and the first term ( a_1 ).Using the formula for the ( n )-th term of an arithmetic sequence:[a_n = a_1 + (n-1)d]I set up two equations based on the given information:[a_3 = a_1 + 2d = 1][a_7 = a_1 + 6d = 3]Subtracting the first equation from the second gives:[4d = 2 implies d = frac{1}{2}]Substituting ( d = frac{1}{2} ) back into the first equation:[a_1 + 2 times frac{1}{2} = 1 implies a_1 + 1 = 1 implies a_1 = 0]Now, I can find the ninth term ( a_9 ):[a_9 = a_1 + 8d = 0 + 8 times frac{1}{2} = 4]Finally, using the sum formula for the first ( n ) terms of an arithmetic sequence:[S_n = frac{n}{2} (a_1 + a_n)]I calculate the sum of the first 9 terms:[S_9 = frac{9}{2} (0 + 4) = frac{9}{2} times 4 = 18]

💡Okay, so I have this geometry problem here: From point ( M ) on the circumcircle of triangle ( ABC ), we drop perpendiculars ( MP ) and ( MQ ) onto lines ( AB ) and ( AC ). I need to figure out for which position of point ( M ) the length of segment ( PQ ) is maximized. Hmm, interesting. Let me try to visualize this first.Alright, triangle ( ABC ) with its circumcircle. Point ( M ) is somewhere on this circle. From ( M ), we drop two perpendiculars: one to ( AB ) which hits at ( P ), and another to ( AC ) which hits at ( Q ). So, ( MP ) is perpendicular to ( AB ), and ( MQ ) is perpendicular to ( AC ). Then, we're supposed to find where ( M ) should be so that the distance between ( P ) and ( Q ) is as long as possible.Let me draw this in my mind. Triangle ( ABC ), circumcircle around it, point ( M ) moving around the circle. As ( M ) moves, points ( P ) and ( Q ) move along ( AB ) and ( AC ) respectively, and ( PQ ) changes length. I need to find the position of ( M ) that makes ( PQ ) the longest.Hmm, okay. Maybe I can use some coordinate geometry here. Let me assign coordinates to the triangle to make it easier. Let's place point ( A ) at the origin, ( (0, 0) ). Let me assume ( AB ) is along the x-axis for simplicity, so point ( B ) is at ( (c, 0) ) for some ( c > 0 ). Point ( C ) can be somewhere in the plane, say ( (d, e) ), but since it's a triangle, ( e ) should not be zero.Wait, but maybe it's better to use a more symmetric approach. Since all points lie on the circumcircle, perhaps using angles and trigonometry would be better. Let me recall that in a circle, the length of a chord is related to the angle it subtends at the center. Maybe that can help.But before that, let me think about the properties of perpendiculars from a point to two sides of a triangle. So, ( MP ) and ( MQ ) are the feet of the perpendiculars from ( M ) to ( AB ) and ( AC ). So, quadrilateral ( APMQ ) has right angles at ( P ) and ( Q ). Is that a cyclic quadrilateral? Wait, no, because only two angles are right angles, not necessarily four.Wait, but maybe I can consider triangles ( APM ) and ( AQM ). Both are right-angled at ( P ) and ( Q ) respectively. So, points ( P ) and ( Q ) lie on the circle with diameter ( AM ). Because in a circle, the right angle subtends the diameter. So, if ( AM ) is the diameter, then any point on the circle will form a right angle with ( A ) and ( M ). So, in this case, since ( MP ) and ( MQ ) are perpendiculars, points ( P ) and ( Q ) lie on the circle with diameter ( AM ).Wait, that seems important. So, both ( P ) and ( Q ) lie on the circle with diameter ( AM ). Therefore, the segment ( PQ ) is a chord of this circle. The length of chord ( PQ ) depends on the angle it subtends at the center of the circle, which is the midpoint of ( AM ).But how does this help me? I need to relate this to the position of ( M ) on the circumcircle of ( ABC ). Maybe I can express ( PQ ) in terms of ( AM ) and some angle.Let me think. Since ( P ) and ( Q ) lie on the circle with diameter ( AM ), the length of ( PQ ) can be expressed as ( PQ = AM cdot sin(theta) ), where ( theta ) is the angle between ( AP ) and ( AQ ). Wait, is that right?Actually, in a circle, the length of a chord is given by ( 2r sin(theta/2) ), where ( r ) is the radius and ( theta ) is the central angle. In this case, the radius is ( AM/2 ), so the length of chord ( PQ ) would be ( 2 cdot (AM/2) cdot sin(theta/2) = AM cdot sin(theta/2) ). Hmm, so maybe my initial thought was a bit off.But what is ( theta ) here? It's the angle between ( AP ) and ( AQ ), which is essentially angle ( PAQ ). But angle ( PAQ ) is the same as angle ( BAC ) because both are the angles at point ( A ) between sides ( AB ) and ( AC ). So, ( theta = angle BAC ).Wait, so if ( PQ = AM cdot sin(angle BAC / 2) ), then to maximize ( PQ ), we need to maximize ( AM ), since ( sin(angle BAC / 2) ) is a constant for a given triangle ( ABC ).But is that correct? Let me double-check. If ( PQ = AM cdot sin(angle BAC / 2) ), then yes, ( PQ ) is directly proportional to ( AM ). So, to maximize ( PQ ), we need to maximize ( AM ).But ( M ) is on the circumcircle of ( ABC ), so ( AM ) is a chord of that circumcircle. The maximum length of chord ( AM ) occurs when ( AM ) is the diameter of the circumcircle. Because the diameter is the longest chord in a circle.Therefore, ( PQ ) is maximized when ( AM ) is the diameter of the circumcircle of ( ABC ). So, point ( M ) must be the point diametrically opposite to ( A ) on the circumcircle.Wait, but in triangle ( ABC ), the circumcircle's diameter opposite to ( A ) would be the point where the altitude from ( A ) meets the circumcircle again, right? Or is it just the reflection of ( A ) over the center?Actually, the diametrically opposite point to ( A ) on the circumcircle is simply the point such that ( AM ) is the diameter. So, if ( O ) is the circumcenter, then ( M ) is the point such that ( O ) is the midpoint of ( AM ).So, in that case, ( M ) is the antipodal point of ( A ) on the circumcircle.Therefore, the position of ( M ) that maximizes ( PQ ) is the point diametrically opposite to ( A ) on the circumcircle of ( ABC ).Let me see if this makes sense. If ( M ) is diametrically opposite to ( A ), then ( AM ) is the diameter, which is the longest possible chord. So, ( AM ) is maximized, which in turn maximizes ( PQ ) because ( PQ ) is proportional to ( AM ).But wait, is ( PQ ) directly proportional to ( AM )? Earlier, I thought ( PQ = AM cdot sin(angle BAC / 2) ). So, if ( angle BAC ) is fixed, then yes, ( PQ ) is directly proportional to ( AM ). So, maximizing ( AM ) would indeed maximize ( PQ ).Alternatively, maybe I can approach this using coordinate geometry to verify.Let me set up a coordinate system with point ( A ) at ( (0, 0) ), ( AB ) along the x-axis, and ( AC ) in the plane. Let me assume specific coordinates for simplicity. Let me take ( AB ) as ( (0, 0) ) to ( (1, 0) ), and ( AC ) as ( (0, 0) ) to ( (0, 1) ). So, triangle ( ABC ) is a right-angled triangle at ( A ).Wait, but in this case, the circumcircle of ( ABC ) would have its diameter as the hypotenuse ( BC ). So, the circumradius is half the length of ( BC ). So, point ( M ) is somewhere on this circle.But in this specific case, since ( ABC ) is right-angled at ( A ), the circumcircle has ( BC ) as its diameter, so the center is the midpoint of ( BC ).So, if I take ( A ) at ( (0, 0) ), ( B ) at ( (1, 0) ), and ( C ) at ( (0, 1) ), then the circumcircle has center at ( (0.5, 0.5) ) and radius ( sqrt{(0.5)^2 + (0.5)^2} = sqrt{0.5} approx 0.7071 ).So, point ( M ) is any point on this circle. Let me parameterize point ( M ) as ( (0.5 + sqrt{0.5} cos theta, 0.5 + sqrt{0.5} sin theta) ).Now, from ( M ), I need to drop perpendiculars to ( AB ) and ( AC ). Since ( AB ) is the x-axis, the foot of the perpendicular from ( M ) to ( AB ) is simply ( (x, 0) ) where ( x ) is the x-coordinate of ( M ). Similarly, the foot of the perpendicular from ( M ) to ( AC ) (which is the y-axis) is ( (0, y) ) where ( y ) is the y-coordinate of ( M ).So, in this coordinate system, point ( P ) is ( (0.5 + sqrt{0.5} cos theta, 0) ) and point ( Q ) is ( (0, 0.5 + sqrt{0.5} sin theta) ).Now, the distance ( PQ ) can be calculated using the distance formula between ( P ) and ( Q ):( PQ = sqrt{(0.5 + sqrt{0.5} cos theta - 0)^2 + (0 - (0.5 + sqrt{0.5} sin theta))^2} )Simplify this:( PQ = sqrt{(0.5 + sqrt{0.5} cos theta)^2 + (-0.5 - sqrt{0.5} sin theta)^2} )Expanding both squares:First term: ( (0.5)^2 + 2 cdot 0.5 cdot sqrt{0.5} cos theta + (sqrt{0.5} cos theta)^2 )= ( 0.25 + sqrt{0.5} cos theta + 0.5 cos^2 theta )Second term: ( (-0.5)^2 + 2 cdot (-0.5) cdot (-sqrt{0.5} sin theta) + (sqrt{0.5} sin theta)^2 )= ( 0.25 + sqrt{0.5} sin theta + 0.5 sin^2 theta )Adding both terms:( 0.25 + sqrt{0.5} cos theta + 0.5 cos^2 theta + 0.25 + sqrt{0.5} sin theta + 0.5 sin^2 theta )= ( 0.5 + sqrt{0.5} (cos theta + sin theta) + 0.5 (cos^2 theta + sin^2 theta) )Since ( cos^2 theta + sin^2 theta = 1 ):= ( 0.5 + sqrt{0.5} (cos theta + sin theta) + 0.5 cdot 1 )= ( 0.5 + sqrt{0.5} (cos theta + sin theta) + 0.5 )= ( 1 + sqrt{0.5} (cos theta + sin theta) )So, ( PQ = sqrt{1 + sqrt{0.5} (cos theta + sin theta)} )Wait, that seems a bit complicated. Maybe I made a mistake in the calculation.Wait, no, actually, the expression inside the square root is:( (0.5 + sqrt{0.5} cos theta)^2 + (0.5 + sqrt{0.5} sin theta)^2 )Wait, no, actually, in my earlier step, I think I messed up the signs. Let me re-examine.Point ( P ) is ( (0.5 + sqrt{0.5} cos theta, 0) ) and point ( Q ) is ( (0, 0.5 + sqrt{0.5} sin theta) ). So, the x-coordinate difference is ( 0.5 + sqrt{0.5} cos theta - 0 = 0.5 + sqrt{0.5} cos theta ), and the y-coordinate difference is ( 0 - (0.5 + sqrt{0.5} sin theta) = -0.5 - sqrt{0.5} sin theta ).So, when squaring, both terms become positive:( (0.5 + sqrt{0.5} cos theta)^2 + (-0.5 - sqrt{0.5} sin theta)^2 )= ( (0.5 + sqrt{0.5} cos theta)^2 + (0.5 + sqrt{0.5} sin theta)^2 )Ah, okay, so both terms are squared, so the negative sign goes away. So, expanding both:First term: ( 0.25 + sqrt{0.5} cos theta + 0.5 cos^2 theta )Second term: ( 0.25 + sqrt{0.5} sin theta + 0.5 sin^2 theta )Adding them together:( 0.25 + 0.25 + sqrt{0.5} (cos theta + sin theta) + 0.5 (cos^2 theta + sin^2 theta) )= ( 0.5 + sqrt{0.5} (cos theta + sin theta) + 0.5 cdot 1 )= ( 0.5 + sqrt{0.5} (cos theta + sin theta) + 0.5 )= ( 1 + sqrt{0.5} (cos theta + sin theta) )So, ( PQ = sqrt{1 + sqrt{0.5} (cos theta + sin theta)} )Hmm, okay. So, to maximize ( PQ ), we need to maximize the expression inside the square root, which is ( 1 + sqrt{0.5} (cos theta + sin theta) ).So, the maximum value occurs when ( cos theta + sin theta ) is maximized. The maximum of ( cos theta + sin theta ) is ( sqrt{2} ), achieved when ( theta = 45^circ ).Wait, so substituting that in, the maximum value inside the square root becomes ( 1 + sqrt{0.5} cdot sqrt{2} = 1 + 1 = 2 ). Therefore, the maximum ( PQ ) is ( sqrt{2} ).But in this specific case, when ( theta = 45^circ ), where is point ( M ) located?Given that ( M ) is parameterized as ( (0.5 + sqrt{0.5} cos theta, 0.5 + sqrt{0.5} sin theta) ), when ( theta = 45^circ ), we have:( x = 0.5 + sqrt{0.5} cdot frac{sqrt{2}}{2} = 0.5 + sqrt{0.5} cdot frac{sqrt{2}}{2} )Simplify ( sqrt{0.5} = frac{sqrt{2}}{2} ), so:( x = 0.5 + frac{sqrt{2}}{2} cdot frac{sqrt{2}}{2} = 0.5 + frac{2}{4} = 0.5 + 0.5 = 1 )Similarly, ( y = 0.5 + sqrt{0.5} cdot frac{sqrt{2}}{2} = 1 )So, point ( M ) is at ( (1, 1) ). But in our coordinate system, the circumcircle has center at ( (0.5, 0.5) ) and radius ( sqrt{0.5} approx 0.7071 ). Wait, but ( (1, 1) ) is at a distance of ( sqrt{(0.5)^2 + (0.5)^2} = sqrt{0.5} ) from the center, so it lies on the circle. Okay, that makes sense.But in this specific case, point ( M ) at ( (1, 1) ) is the point diametrically opposite to ( A ) at ( (0, 0) ). Because the center is at ( (0.5, 0.5) ), so the diametrically opposite point to ( (0, 0) ) is ( (1, 1) ).So, in this coordinate system, the maximum ( PQ ) occurs when ( M ) is diametrically opposite to ( A ). That seems consistent with my earlier conclusion.But wait, in this case, triangle ( ABC ) was right-angled at ( A ). Does this result hold for any triangle ( ABC ), or is it specific to right-angled triangles?Let me think. In a general triangle, the circumcircle is not necessarily having ( BC ) as the diameter unless it's a right-angled triangle. So, in a general triangle, the diametrically opposite point to ( A ) is not necessarily the midpoint of ( BC ) or anything like that. It's just another point on the circumcircle.But in the specific case of a right-angled triangle, the diametrically opposite point to ( A ) is indeed the midpoint of the hypotenuse, which is the circumradius. So, in that case, the point ( M ) is the midpoint of ( BC ).But in a general triangle, the point diametrically opposite to ( A ) is just another point on the circumcircle, not necessarily related to the sides ( AB ) and ( AC ).But in both cases, whether the triangle is right-angled or not, the maximum ( PQ ) occurs when ( M ) is diametrically opposite to ( A ). Because in the coordinate system, that's when ( AM ) is the diameter, which is the longest possible chord, and hence, ( PQ ) is maximized.Wait, but in the coordinate system, when ( M ) is diametrically opposite to ( A ), ( AM ) is the diameter, so ( AM = 2R ), where ( R ) is the circumradius. So, in that case, ( PQ = AM cdot sin(angle BAC / 2) = 2R cdot sin(angle BAC / 2) ).But in the specific case of the right-angled triangle, ( angle BAC = 90^circ ), so ( sin(45^circ) = frac{sqrt{2}}{2} ), so ( PQ = 2R cdot frac{sqrt{2}}{2} = R sqrt{2} ). In our coordinate system, ( R = sqrt{0.5} ), so ( PQ = sqrt{0.5} cdot sqrt{2} = 1 ). Wait, but earlier, we found ( PQ = sqrt{2} ). Hmm, that seems inconsistent.Wait, no, in the coordinate system, the maximum ( PQ ) was ( sqrt{2} ), but according to this formula, it should be ( R sqrt{2} = sqrt{0.5} cdot sqrt{2} = 1 ). There's a discrepancy here.Wait, maybe my formula was wrong. Let me re-examine.Earlier, I thought ( PQ = AM cdot sin(angle BAC / 2) ), but in the coordinate system, when ( AM ) is the diameter, ( AM = 2R = 2 cdot sqrt{0.5} = sqrt{2} ). Then, ( PQ = sqrt{2} cdot sin(45^circ) = sqrt{2} cdot frac{sqrt{2}}{2} = 1 ). But in reality, in the coordinate system, ( PQ ) was ( sqrt{2} ). So, my formula must be incorrect.Wait, perhaps I confused the angle. Maybe ( PQ = AM cdot sin(angle BAC) ) instead of half the angle.Let me check. If ( PQ = AM cdot sin(angle BAC) ), then in the right-angled triangle, ( angle BAC = 90^circ ), so ( sin(90^circ) = 1 ), so ( PQ = AM cdot 1 = AM ). But in our coordinate system, ( AM ) when ( M ) is diametrically opposite is ( sqrt{(1)^2 + (1)^2} = sqrt{2} ), which matches the calculated ( PQ = sqrt{2} ). So, perhaps my initial formula was wrong, and it should be ( PQ = AM cdot sin(angle BAC) ).Wait, let me think again. In the circle with diameter ( AM ), points ( P ) and ( Q ) lie on this circle. The angle ( PAQ ) is equal to ( angle BAC ). So, in the circle with diameter ( AM ), the chord ( PQ ) subtends an angle of ( angle PAQ = angle BAC ) at point ( A ). Therefore, the length of chord ( PQ ) is ( 2r sin(theta / 2) ), where ( r ) is the radius of the circle, which is ( AM / 2 ), and ( theta ) is ( angle PAQ = angle BAC ).So, ( PQ = 2 cdot (AM / 2) cdot sin(angle BAC / 2) = AM cdot sin(angle BAC / 2) ).But in our coordinate system, when ( M ) is diametrically opposite, ( AM = sqrt{2} ), and ( angle BAC = 90^circ ), so ( PQ = sqrt{2} cdot sin(45^circ) = sqrt{2} cdot frac{sqrt{2}}{2} = 1 ). But in reality, we found ( PQ = sqrt{2} ). So, something is wrong here.Wait, maybe the angle subtended by ( PQ ) at the center is not ( angle BAC ), but something else. Let me think.In the circle with diameter ( AM ), the points ( P ) and ( Q ) are the feet of the perpendiculars from ( M ) to ( AB ) and ( AC ). So, the angle ( PAQ ) is indeed ( angle BAC ), but in the circle with diameter ( AM ), the central angle subtended by ( PQ ) is ( 2 cdot angle PAQ ), because the central angle is twice the inscribed angle.Wait, that might be the case. So, if ( angle PAQ = angle BAC ), then the central angle is ( 2 cdot angle BAC ), and hence, the length of chord ( PQ ) is ( 2r sin(angle BAC) ), where ( r = AM / 2 ).So, ( PQ = 2 cdot (AM / 2) cdot sin(angle BAC) = AM cdot sin(angle BAC) ).In our coordinate system, ( AM = sqrt{2} ), ( angle BAC = 90^circ ), so ( PQ = sqrt{2} cdot 1 = sqrt{2} ), which matches our earlier calculation. So, the correct formula is ( PQ = AM cdot sin(angle BAC) ).Therefore, to maximize ( PQ ), we need to maximize ( AM ), since ( sin(angle BAC) ) is a constant for a given triangle. The maximum value of ( AM ) occurs when ( AM ) is the diameter of the circumcircle. Hence, ( PQ ) is maximized when ( M ) is the point diametrically opposite to ( A ) on the circumcircle.This seems consistent now. So, in both the general case and the specific coordinate system, the conclusion is the same: ( PQ ) is maximized when ( M ) is diametrically opposite to ( A ).Therefore, the position of ( M ) that maximizes ( PQ ) is the point diametrically opposite to ( A ) on the circumcircle of triangle ( ABC ).**Final Answer**The length of segment ( PQ ) is maximized when ( M ) is the point diametrically opposite to ( A ) on the circumcircle of triangle ( ABC ). Thus, the answer is (boxed{M text{ is the antipodal point of } A}).**Final Answer**The length of segment ( PQ ) is maximized when ( M ) is the point diametrically opposite to ( A ) on the circumcircle of triangle ( ABC ). Thus, the answer is (boxed{M text{ is the antipodal point of } A}).

💡First, I need to understand the set operation defined as ( A * B = { z mid z = xy, x in A, y in B } ). This means that for every element ( x ) in set ( A ) and every element ( y ) in set ( B ), I multiply them together to form the elements of the new set ( A * B ).Given the sets ( A = {1, 2} ) and ( B = {0, 2} ), I will calculate all possible products of elements from ( A ) and ( B ).1. Multiply each element of ( A ) by each element of ( B ): - ( 1 times 0 = 0 ) - ( 1 times 2 = 2 ) - ( 2 times 0 = 0 ) - ( 2 times 2 = 4 )2. Collect the results into the set ( A * B ), ensuring that each element is unique: - ( A * B = {0, 2, 4} )3. Calculate the sum of all elements in the set ( A * B ): - ( 0 + 2 + 4 = 6 )Therefore, the sum of all elements in the set ( A * B ) is 6.

💡Alright, so I have this problem here: I need to insert a number between 3 and an unknown number to form an arithmetic sequence. Then, if I reduce the middle term by 6, the sequence becomes a geometric sequence. I need to find this unknown number. Hmm, okay, let's break this down step by step.First, let's recall what an arithmetic sequence is. It's a sequence where the difference between consecutive terms is constant. So, if I have three numbers in an arithmetic sequence, the middle number is the average of the first and the third. That makes sense. So, if I have numbers 3, a, and b in an arithmetic sequence, the difference between 3 and a should be the same as the difference between a and b.Mathematically, that would mean:a - 3 = b - aSo, if I solve this equation, I can express a in terms of b. Let's do that:a - 3 = b - aAdding a to both sides:2a - 3 = bThen, adding 3 to both sides:2a = b + 3So, a = (b + 3)/2Okay, so now I have a in terms of b. That's good progress.Next, the problem says that if the middle term is reduced by 6, the sequence becomes a geometric sequence. A geometric sequence is one where each term is a multiple of the previous term by a common ratio. So, if I reduce the middle term a by 6, the new sequence becomes 3, a - 6, b, and this should be a geometric sequence.In a geometric sequence, the ratio between consecutive terms is constant. So, the ratio of the second term to the first term should be equal to the ratio of the third term to the second term. That gives us:(a - 6)/3 = b/(a - 6)So, now I have another equation involving a and b. Since I already have a expressed in terms of b from the arithmetic sequence, I can substitute that into this equation.Let me write that substitution out:a = (b + 3)/2So, a - 6 = (b + 3)/2 - 6Let me simplify that:a - 6 = (b + 3)/2 - 6To combine these terms, I'll get a common denominator:= (b + 3)/2 - 12/2= (b + 3 - 12)/2= (b - 9)/2Okay, so a - 6 simplifies to (b - 9)/2.Now, plug this back into the geometric sequence ratio equation:(a - 6)/3 = b/(a - 6)Substituting a - 6:[(b - 9)/2]/3 = b/[(b - 9)/2]Simplify the left side:= (b - 9)/6Simplify the right side:= b / [(b - 9)/2] = b * (2/(b - 9)) = 2b/(b - 9)So now, the equation becomes:(b - 9)/6 = 2b/(b - 9)Hmm, okay, so now I have:(b - 9)/6 = 2b/(b - 9)To solve this equation, I can cross-multiply:(b - 9)^2 = 6 * 2b(b - 9)^2 = 12bLet me expand the left side:b^2 - 18b + 81 = 12bNow, bring all terms to one side:b^2 - 18b + 81 - 12b = 0b^2 - 30b + 81 = 0Wait, that doesn't seem right. Let me double-check my steps.Wait, when I cross-multiplied, I had (b - 9)^2 = 12b. Expanding (b - 9)^2 gives b^2 - 18b + 81. So, bringing 12b to the left side:b^2 - 18b + 81 - 12b = 0Which simplifies to:b^2 - 30b + 81 = 0Wait, that seems correct. But let me check my earlier steps again to make sure I didn't make a mistake.Starting from the geometric sequence ratio:(a - 6)/3 = b/(a - 6)Substituting a = (b + 3)/2:[(b + 3)/2 - 6]/3 = b/[(b + 3)/2 - 6]Simplify numerator:(b + 3)/2 - 6 = (b + 3 - 12)/2 = (b - 9)/2So, left side becomes:[(b - 9)/2]/3 = (b - 9)/6Right side:b / [(b - 9)/2] = 2b/(b - 9)So, equation is:(b - 9)/6 = 2b/(b - 9)Cross-multiplying:(b - 9)^2 = 12bYes, that's correct. So, expanding:b^2 - 18b + 81 = 12bBring all terms to left:b^2 - 18b + 81 - 12b = 0b^2 - 30b + 81 = 0Now, let's solve this quadratic equation. I can use the quadratic formula:b = [30 ± sqrt(900 - 324)]/2= [30 ± sqrt(576)]/2= [30 ± 24]/2So, two solutions:b = (30 + 24)/2 = 54/2 = 27b = (30 - 24)/2 = 6/2 = 3Wait, so b could be 27 or 3. But b is the unknown number, and we started with 3 as the first term. If b is 3, then the sequence would be 3, a, 3. But that would mean a is also 3, which doesn't make sense because we're inserting a number between 3 and b, implying that b should be different from 3. So, b = 3 is not a valid solution in this context.Therefore, the only valid solution is b = 27.Let me verify this. If b = 27, then a = (27 + 3)/2 = 30/2 = 15.So, the arithmetic sequence is 3, 15, 27. The common difference is 12, which checks out.Now, reducing the middle term by 6 gives us 15 - 6 = 9. So, the new sequence is 3, 9, 27.Is this a geometric sequence? Let's check the ratios:9/3 = 327/9 = 3Yes, the common ratio is 3, so it is indeed a geometric sequence.Therefore, the unknown number b is 27.

💡Okay, so I have this quadratic inequality: ( ax^2 - (1 + a)x + b geq 0 ). The solution set is given as ( {x | -frac{1}{5} leq x leq 1} ). I need to find the value of ( a + b ). Hmm, let me think about how to approach this.First, I remember that when a quadratic inequality has a solution set between two values, like ( -frac{1}{5} ) and ( 1 ), it means that the quadratic opens downwards. So, the coefficient of ( x^2 ), which is ( a ), must be negative. That's important because it tells me about the direction of the parabola.Next, since the solution set is between ( -frac{1}{5} ) and ( 1 ), these must be the roots of the quadratic equation ( ax^2 - (1 + a)x + b = 0 ). So, the roots are ( x = -frac{1}{5} ) and ( x = 1 ).I recall that for a quadratic equation ( ax^2 + bx + c = 0 ), the sum of the roots is ( -b/a ) and the product is ( c/a ). But in this case, the quadratic is ( ax^2 - (1 + a)x + b ), so let me adjust the formula accordingly.Let me denote the roots as ( r_1 = -frac{1}{5} ) and ( r_2 = 1 ). Then, the sum of the roots ( r_1 + r_2 ) should be equal to ( frac{(1 + a)}{a} ) because the coefficient of ( x ) is ( -(1 + a) ). Similarly, the product of the roots ( r_1 times r_2 ) should be equal to ( frac{b}{a} ).So, let me write down these equations:1. Sum of roots: ( r_1 + r_2 = frac{1 + a}{a} )2. Product of roots: ( r_1 times r_2 = frac{b}{a} )Plugging in the values:1. ( -frac{1}{5} + 1 = frac{1 + a}{a} )2. ( -frac{1}{5} times 1 = frac{b}{a} )Let me simplify the first equation:( -frac{1}{5} + 1 = frac{1 + a}{a} )Calculating the left side:( -frac{1}{5} + 1 = frac{-1 + 5}{5} = frac{4}{5} )So, ( frac{4}{5} = frac{1 + a}{a} )Let me solve for ( a ):Multiply both sides by ( a ):( frac{4}{5}a = 1 + a )Subtract ( a ) from both sides:( frac{4}{5}a - a = 1 )Convert ( a ) to fifths:( frac{4}{5}a - frac{5}{5}a = 1 )Which simplifies to:( -frac{1}{5}a = 1 )Multiply both sides by ( -5 ):( a = -5 )Okay, so ( a = -5 ). Now, let's find ( b ) using the product of the roots.From the second equation:( -frac{1}{5} times 1 = frac{b}{a} )We know ( a = -5 ), so:( -frac{1}{5} = frac{b}{-5} )Multiply both sides by ( -5 ):( -frac{1}{5} times (-5) = b )Which simplifies to:( 1 = b )So, ( b = 1 ).Now, the question asks for ( a + b ):( a + b = -5 + 1 = -4 )Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting with the sum of roots:( -frac{1}{5} + 1 = frac{4}{5} ), which equals ( frac{1 + a}{a} ). Plugging ( a = -5 ):( frac{1 + (-5)}{-5} = frac{-4}{-5} = frac{4}{5} ). That checks out.For the product:( -frac{1}{5} times 1 = -frac{1}{5} ), which equals ( frac{b}{a} = frac{1}{-5} = -frac{1}{5} ). That also checks out.So, ( a = -5 ) and ( b = 1 ), so ( a + b = -4 ). Looking at the options, that's option C.I think that's correct. I don't see any errors in my reasoning.**Final Answer**boxed{C}

💡First, determine the number of words Leo can write on one page by multiplying the number of lines per page by the number of words per line.Next, calculate the total number of words Leo has already written by multiplying the number of pages he has filled by the number of words per page.Finally, subtract the number of words Leo has already written from the total number of words required to find out how many words he still needs to write.