💡Okay, so I have this problem about a sequence {a_n} where each term is given by a_n = n² + λn. The question is asking for the range of values for the real number λ such that the sequence is increasing. Hmm, okay. Let me try to figure this out step by step.First, I remember that for a sequence to be increasing, each term must be greater than the previous one. So, that means a_{n+1} should be greater than a_n for all n. In other words, the difference a_{n+1} - a_n should be positive for all n.Let me write that down. So, a_{n+1} - a_n > 0 for all n ≥ 1.Given that a_n = n² + λn, let's compute a_{n+1}:a_{n+1} = (n+1)² + λ(n+1)Expanding that, (n+1)² is n² + 2n + 1, and λ(n+1) is λn + λ. So, putting it all together:a_{n+1} = n² + 2n + 1 + λn + λNow, let's subtract a_n from a_{n+1}:a_{n+1} - a_n = (n² + 2n + 1 + λn + λ) - (n² + λn)Simplify that:= n² + 2n + 1 + λn + λ - n² - λnThe n² terms cancel out, and the λn terms also cancel out. So we're left with:= 2n + 1 + λSo, the difference a_{n+1} - a_n simplifies to 2n + 1 + λ.Since the sequence is increasing, this difference must be greater than zero for all n ≥ 1. So, we have:2n + 1 + λ > 0 for all n ≥ 1.Now, I need to find the values of λ that satisfy this inequality for all n starting from 1.Let me think about this. Since n starts at 1 and increases, the expression 2n + 1 + λ is a linear function in terms of n. As n increases, 2n + 1 + λ will also increase because the coefficient of n is positive (2). So, the smallest value of this expression occurs when n is smallest, which is n = 1.Therefore, if I can ensure that the expression is positive when n = 1, it will automatically be positive for all larger n. That makes sense because as n increases, the expression only gets larger.So, let's plug n = 1 into the expression:2(1) + 1 + λ > 0Simplify that:2 + 1 + λ > 0Which is:3 + λ > 0Subtracting 3 from both sides:λ > -3So, this tells me that λ must be greater than -3 for the sequence to be increasing.Wait, let me double-check that. If λ is greater than -3, then for n = 1, the difference is 3 + λ, which is positive. For n = 2, the difference is 5 + λ, which is even larger, so it will still be positive as long as λ > -3. Similarly, for n = 3, it's 7 + λ, which is even more positive. So, yeah, it seems that as long as λ is greater than -3, the difference will always be positive for all n ≥ 1.Is there any other consideration I need to make? Let me think. The sequence is defined for n = 1, 2, 3, and so on. So, I don't have to worry about n being zero or negative because n starts at 1. Therefore, my condition is solely based on n starting at 1.Another way to think about it is that the sequence is increasing if each term is larger than the one before it. The difference between consecutive terms is 2n + 1 + λ. Since n is always at least 1, the smallest difference occurs at n = 1, which is 3 + λ. So, to ensure that even the smallest difference is positive, we set 3 + λ > 0, leading to λ > -3.I think that's solid. So, the range of λ is all real numbers greater than -3.**Final Answer**The range of values for the real number λ is boxed{(-3, +infty)}.

💡Okay, so I have this problem where I need to find the maximum value of the expression:[ A = frac{left(x^{2} - yright) sqrt{y + x^{3} - x y} + left(y^{2} - xright) sqrt{x + y^{3} - x y} + 1}{(x - y)^{2} + 1} ]for ( x, y ) in the interval ( (0,1] ). Hmm, this looks a bit complicated, but let me try to break it down step by step.First, I notice that the denominator is ( (x - y)^2 + 1 ). Since squares are always non-negative, the smallest value the denominator can take is 1 (when ( x = y )), and it increases as ( x ) and ( y ) get further apart. So, the denominator is always between 1 and some larger value depending on how different ( x ) and ( y ) are.Now, looking at the numerator, it has two main parts:1. ( (x^{2} - y) sqrt{y + x^{3} - x y} )2. ( (y^{2} - x) sqrt{x + y^{3} - x y} )Plus 1 at the end. Both of these terms involve square roots, which makes me think about the expressions inside the square roots. For the square roots to be real numbers, the expressions inside must be non-negative. So, I should check when ( y + x^{3} - x y geq 0 ) and ( x + y^{3} - x y geq 0 ).Let me simplify these expressions:For the first square root:[ y + x^{3} - x y = y(1 - x) + x^{3} ]For the second square root:[ x + y^{3} - x y = x(1 - y) + y^{3} ]Since ( x, y in (0,1] ), both ( 1 - x ) and ( 1 - y ) are non-negative. Therefore, both expressions inside the square roots are non-negative, so the square roots are real and defined for all ( x, y ) in the given interval.Now, let me think about the terms ( (x^{2} - y) ) and ( (y^{2} - x) ). Depending on whether ( x^2 ) is greater than or less than ( y ), and similarly for ( y^2 ) and ( x ), these terms can be positive or negative.Let me consider two cases:**Case 1: ( x^2 geq y ) and ( y^2 geq x )**In this case, both ( (x^{2} - y) ) and ( (y^{2} - x) ) are non-negative. So, both terms in the numerator are non-negative, and adding 1 will make the numerator larger. Since the denominator is at least 1, the whole expression ( A ) could be larger than 1.But wait, I need to check if this case is possible. If ( x^2 geq y ) and ( y^2 geq x ), does that hold for some ( x, y ) in ( (0,1] )?Let me test with ( x = y = 1 ). Then, ( x^2 = 1 geq y = 1 ) and ( y^2 = 1 geq x = 1 ). So, equality holds here.What about ( x = y = frac{1}{2} )? Then, ( x^2 = frac{1}{4} ) which is less than ( y = frac{1}{2} ), so this case doesn't hold. So, it seems that this case only holds when ( x ) and ( y ) are close to 1.**Case 2: ( x^2 < y ) or ( y^2 < x )**In this case, at least one of ( (x^{2} - y) ) or ( (y^{2} - x) ) is negative. So, the corresponding term in the numerator would be negative, which might reduce the overall value of ( A ).But since both terms are multiplied by square roots, which are positive, the sign of each term depends on ( (x^{2} - y) ) and ( (y^{2} - x) ).I think it's useful to consider symmetry here. If I swap ( x ) and ( y ), the expression remains the same. So, without loss of generality, I can assume ( x geq y ) and analyze accordingly.But before that, let me see if I can simplify the expression further.Looking at the numerator:[ (x^{2} - y) sqrt{y + x^{3} - x y} + (y^{2} - x) sqrt{x + y^{3} - x y} + 1 ]I wonder if there's a way to bound this expression. Maybe I can find an upper bound for the numerator and a lower bound for the denominator to find an upper bound for ( A ).Let me consider the terms inside the square roots again:For ( sqrt{y + x^{3} - x y} ), as I simplified earlier, it's ( sqrt{y(1 - x) + x^{3}} ). Since ( y leq 1 ) and ( x leq 1 ), ( y(1 - x) leq 1 - x ) and ( x^{3} leq x ). So, ( y + x^{3} - x y leq (1 - x) + x = 1 ). Therefore, ( sqrt{y + x^{3} - x y} leq 1 ).Similarly, ( sqrt{x + y^{3} - x y} leq 1 ).So, both square roots are bounded above by 1. That might help in bounding the numerator.Now, let's consider the terms ( (x^{2} - y) ) and ( (y^{2} - x) ). Since ( x, y in (0,1] ), ( x^2 leq x ) and ( y^2 leq y ). So, ( x^{2} - y leq x - y ) and ( y^{2} - x leq y - x ).Therefore, ( (x^{2} - y) leq x - y ) and ( (y^{2} - x) leq y - x ).So, the numerator can be bounded as:[ (x^{2} - y) sqrt{y + x^{3} - x y} + (y^{2} - x) sqrt{x + y^{3} - x y} + 1 leq (x - y) cdot 1 + (y - x) cdot 1 + 1 = 0 + 1 = 1 ]Wait, that seems interesting. So, the numerator is bounded above by 1, and the denominator is at least 1. Therefore, ( A leq 1 ).But is this tight? When does equality hold?For equality to hold in the numerator bound, we need:1. ( (x^{2} - y) sqrt{y + x^{3} - x y} = x - y )2. ( (y^{2} - x) sqrt{x + y^{3} - x y} = y - x )But wait, ( (x^{2} - y) leq x - y ) and ( (y^{2} - x) leq y - x ). So, for equality, we need ( x^{2} - y = x - y ) and ( y^{2} - x = y - x ).Simplifying ( x^{2} - y = x - y ), we get ( x^{2} = x ), which implies ( x = 0 ) or ( x = 1 ). But ( x in (0,1] ), so ( x = 1 ).Similarly, ( y^{2} - x = y - x ) implies ( y^{2} = y ), so ( y = 0 ) or ( y = 1 ). Again, ( y = 1 ).Therefore, equality holds when ( x = y = 1 ).Let me check this:Plugging ( x = y = 1 ) into the original expression:Numerator:[ (1^2 - 1) sqrt{1 + 1^3 - 1 cdot 1} + (1^2 - 1) sqrt{1 + 1^3 - 1 cdot 1} + 1 = (0) cdot sqrt{1 + 1 - 1} + (0) cdot sqrt{1 + 1 - 1} + 1 = 0 + 0 + 1 = 1 ]Denominator:[ (1 - 1)^2 + 1 = 0 + 1 = 1 ]So, ( A = 1/1 = 1 ).Therefore, the maximum value of ( A ) is indeed 1, achieved when ( x = y = 1 ).But just to be thorough, let me check another point to see if ( A ) can be greater than 1.Let me try ( x = 1 ) and ( y = 0.5 ):Numerator:[ (1^2 - 0.5) sqrt{0.5 + 1^3 - 1 cdot 0.5} + (0.5^2 - 1) sqrt{1 + 0.5^3 - 1 cdot 0.5} + 1 ][ = (0.5) sqrt{0.5 + 1 - 0.5} + (-0.75) sqrt{1 + 0.125 - 0.5} + 1 ][ = 0.5 sqrt{1} + (-0.75) sqrt{0.625} + 1 ][ = 0.5 cdot 1 + (-0.75) cdot 0.7906 + 1 ][ = 0.5 - 0.5929 + 1 ][ = 0.5 - 0.5929 + 1 = 0.9071 ]Denominator:[ (1 - 0.5)^2 + 1 = 0.25 + 1 = 1.25 ]So, ( A = 0.9071 / 1.25 approx 0.7257 ), which is less than 1.What about ( x = 0.5 ) and ( y = 0.5 ):Numerator:[ (0.25 - 0.5) sqrt{0.5 + 0.125 - 0.25} + (0.25 - 0.5) sqrt{0.5 + 0.125 - 0.25} + 1 ][ = (-0.25) sqrt{0.375} + (-0.25) sqrt{0.375} + 1 ][ = -0.25 cdot 0.6124 - 0.25 cdot 0.6124 + 1 ][ = -0.1531 - 0.1531 + 1 ][ = -0.3062 + 1 = 0.6938 ]Denominator:[ (0.5 - 0.5)^2 + 1 = 0 + 1 = 1 ]So, ( A = 0.6938 / 1 = 0.6938 ), which is still less than 1.Another test case: ( x = 1 ), ( y = 0.9 ):Numerator:[ (1 - 0.9) sqrt{0.9 + 1 - 0.9} + (0.81 - 1) sqrt{1 + 0.729 - 0.9} + 1 ][ = 0.1 sqrt{1} + (-0.19) sqrt{0.829} + 1 ][ = 0.1 cdot 1 + (-0.19) cdot 0.9105 + 1 ][ = 0.1 - 0.173 + 1 ][ = 0.927 ]Denominator:[ (1 - 0.9)^2 + 1 = 0.01 + 1 = 1.01 ]So, ( A = 0.927 / 1.01 approx 0.9178 ), still less than 1.What if I take ( x = 1 ) and ( y ) approaching 1? Let's say ( y = 1 - epsilon ) where ( epsilon ) is very small.Numerator:[ (1 - (1 - epsilon)) sqrt{(1 - epsilon) + 1 - 1 cdot (1 - epsilon)} + ((1 - epsilon)^2 - 1) sqrt{1 + (1 - epsilon)^3 - 1 cdot (1 - epsilon)} + 1 ]Simplify:[ (epsilon) sqrt{(1 - epsilon) + 1 - (1 - epsilon)} + ((1 - 2epsilon + epsilon^2) - 1) sqrt{1 + (1 - 3epsilon + 3epsilon^2 - epsilon^3) - (1 - epsilon)} + 1 ][ = epsilon sqrt{1} + (-2epsilon + epsilon^2) sqrt{1 + 1 - 3epsilon + 3epsilon^2 - epsilon^3 - 1 + epsilon} + 1 ][ = epsilon + (-2epsilon + epsilon^2) sqrt{1 - 2epsilon + 3epsilon^2 - epsilon^3} + 1 ]As ( epsilon ) approaches 0, the square root term approaches 1, so:[ approx epsilon + (-2epsilon) cdot 1 + 1 = -epsilon + 1 ]Denominator:[ (1 - (1 - epsilon))^2 + 1 = epsilon^2 + 1 ]So, as ( epsilon ) approaches 0, ( A ) approaches ( (1 - epsilon) / (1 + epsilon^2) approx 1 - epsilon ), which is slightly less than 1.This suggests that as ( y ) approaches 1 from below, ( A ) approaches 1 but doesn't exceed it.What about ( x ) and ( y ) both approaching 1? Let me set ( x = 1 - epsilon ) and ( y = 1 - delta ) with ( epsilon, delta ) small.But this might get too complicated. Instead, let me consider the case when ( x = y ). If ( x = y ), then the expression simplifies.Let ( x = y ). Then, the expression becomes:Numerator:[ (x^2 - x) sqrt{x + x^3 - x^2} + (x^2 - x) sqrt{x + x^3 - x^2} + 1 ][ = 2(x^2 - x) sqrt{x + x^3 - x^2} + 1 ]Denominator:[ (x - x)^2 + 1 = 0 + 1 = 1 ]So, ( A = 2(x^2 - x) sqrt{x + x^3 - x^2} + 1 )Let me simplify the expression inside the square root:[ x + x^3 - x^2 = x(1 - x + x^2) ]Since ( x in (0,1] ), ( 1 - x + x^2 ) is always positive (as the discriminant of ( x^2 - x + 1 ) is negative), so the square root is real.Now, let me analyze ( A ) when ( x = y ):[ A = 2(x^2 - x) sqrt{x(1 - x + x^2)} + 1 ]Note that ( x^2 - x = x(x - 1) ), which is negative for ( x in (0,1) ). So, ( A ) becomes:[ A = 2 cdot text{negative} cdot sqrt{text{positive}} + 1 ]Which is less than 1. So, when ( x = y ), ( A ) is less than 1, except when ( x = y = 1 ), where ( A = 1 ).Therefore, the maximum value of ( A ) is achieved when ( x = y = 1 ), giving ( A = 1 ).I think I've covered different cases and tested specific points, and in all cases, ( A ) doesn't exceed 1. So, I'm pretty confident that the maximum value of ( A ) is indeed 1.

💡Okay, so I have this problem about a pentagonal pyramid. Let me try to visualize it first. A pentagonal pyramid has a base that's a pentagon, and then five triangular faces that come up to meet at a single point, which is the apex of the pyramid. So, in total, there are six faces: one pentagonal base and five triangular sides.The problem says that the total surface area is 30, and the lateral surface area is 25. I need to find the sum of the areas of the two bases. Wait, two bases? Hmm, usually a pyramid has just one base, right? So, maybe this is a double pyramid or something? Or perhaps it's a typo, and they meant the sum of the areas of the base and the top, but in a pyramid, the top is just the apex, which doesn't have an area. That doesn't make sense.Wait, maybe it's a different kind of pyramid. Let me think. Oh, maybe it's a bipyramid, which has two pentagonal bases connected by triangles. So, like two pentagonal pyramids stuck together at their bases. That would make sense because then there are two bases. So, if that's the case, the total surface area would include both pentagonal bases and all the triangular faces.But the problem mentions it's a pentagonal pyramid, not a bipyramid. Hmm, maybe I'm overcomplicating it. Let's read the problem again carefully."The surface area of a pentagonal pyramid ABCDE-A₁B₁C₁D₁E₁ is 30, and the lateral surface area equals 25. Then, the sum of the areas of the two bases is..."Wait, the notation ABCDE-A₁B₁C₁D₁E₁ suggests that there are two pentagons: one labeled ABCDE and another labeled A₁B₁C₁D₁E₁. So, maybe it's a prism instead of a pyramid? Because a prism has two congruent bases connected by rectangles. But the problem says it's a pyramid, so that's confusing.Alternatively, maybe it's a pyramid with two different bases? But pyramids typically have one base and triangular faces meeting at an apex. So, perhaps the problem is referring to the base and the top face, but in a pyramid, the top is just a point, not a face with area.Wait, maybe it's a frustum of a pyramid, which has two parallel bases, both pentagons, and trapezoidal lateral faces. But the problem says it's a pyramid, not a frustum.This is getting confusing. Let me try to parse the problem again.Total surface area is 30, lateral surface area is 25. So, total surface area includes all faces, and lateral surface area includes only the sides, not the bases. So, if I subtract the lateral surface area from the total surface area, I should get the sum of the areas of the bases.But the problem says "the sum of the areas of the two bases." So, if it's a pyramid, there should be only one base. Unless it's a bipyramid, which has two bases. So, maybe it's a bipyramid.In a bipyramid, you have two pentagonal bases and a set of triangular faces connecting them. So, the total surface area would be the sum of the areas of the two pentagons plus the areas of all the triangular faces.Given that, if the total surface area is 30 and the lateral surface area (which would be the triangular faces) is 25, then the sum of the areas of the two bases would be 30 minus 25, which is 5. So, the answer would be 5, which is option A.But wait, the problem says it's a pentagonal pyramid, not a bipyramid. So, maybe I'm misinterpreting it. Let me think again.If it's a pentagonal pyramid, there is one pentagonal base and five triangular faces. So, the total surface area is the area of the base plus the areas of the five triangles. The lateral surface area is just the areas of the five triangles, which is 25. So, if the total surface area is 30, then the area of the base would be 30 minus 25, which is 5. But the problem asks for the sum of the areas of the two bases. Wait, there's only one base in a pyramid. So, maybe the problem is referring to the base and the top face, but in a pyramid, the top is just a point, not a face with area.Alternatively, maybe it's a typo, and they meant the sum of the areas of the base and the lateral faces, but that would be the total surface area, which is already given as 30.Wait, perhaps the pyramid is truncated, making it a frustum, which has two pentagonal bases and five trapezoidal faces. In that case, the lateral surface area would be the sum of the areas of the trapezoids, and the total surface area would include both pentagons and the trapezoids. So, if the total surface area is 30 and the lateral surface area is 25, then the sum of the areas of the two bases would be 30 minus 25, which is 5. So, again, the answer would be 5.But the problem specifically says it's a pentagonal pyramid, not a frustum. So, I'm not sure. Maybe the problem is referring to the base and the top face, but in a pyramid, the top face is just a point, so its area is zero. That wouldn't make sense.Alternatively, maybe it's a double pyramid, meaning two pyramids stuck together at their bases, forming a bipyramid. In that case, there are two pentagonal bases and ten triangular faces (five from each pyramid). But the problem mentions only one set of vertices: ABCDE-A₁B₁C₁D₁E₁. So, maybe it's a single pyramid with two sets of vertices, but that seems unlikely.Wait, maybe the notation ABCDE-A₁B₁C₁D₁E₁ refers to the base ABCDE and the apex A₁B₁C₁D₁E₁, but that doesn't make sense because the apex should be a single point, not another pentagon.This is getting too confusing. Let me try to approach it differently.Total surface area = 30Lateral surface area = 25Sum of the areas of the two bases = ?If it's a pyramid, total surface area = base area + lateral surface areaSo, base area = total surface area - lateral surface area = 30 - 25 = 5But the problem asks for the sum of the areas of the two bases. If it's a pyramid, there's only one base, so the sum would be 5. But if it's a bipyramid, there are two bases, each with area 5/2, so the sum would still be 5.Alternatively, if it's a frustum, the sum of the two bases would be 5.So, regardless of whether it's a pyramid, bipyramid, or frustum, the sum of the areas of the two bases would be 5.Therefore, the answer is 5, which is option A.

💡First, I need to understand the relationship between Alyosha's floor number and Ira's apartment number. Since they are equal, I'll denote this common value as ( K ).Next, I know that each floor has 9 apartments. This means the apartment numbers on the ( K )-th floor range from ( 9K - 8 ) to ( 9K ).Let ( A ) represent Alyosha's apartment number. According to the problem, the sum of Alyosha's and Ira's apartment numbers is 329, so:[A + K = 329]This implies:[A = 329 - K]Since Alyosha's apartment number must be within the range of the ( K )-th floor, I set up the inequality:[9K - 8 leq A leq 9K]Substituting ( A = 329 - K ) into the inequality:[9K - 8 leq 329 - K leq 9K]Solving the lower bound:[9K - 8 leq 329 - K][10K leq 337][K leq 33.7]Solving the upper bound:[329 - K leq 9K][329 leq 10K][K geq 32.9]Combining both bounds, ( K ) must be between 32.9 and 33.7. Since ( K ) must be an integer, the only valid value is:[K = 33]Finally, substituting ( K = 33 ) back into the equation for ( A ):[A = 329 - 33 = 296]

💡First, I need to determine the number of Rs. 50 notes Kiran has. Since the total amount in Rs. 50 notes is Rs. 3500, I can find the number of these notes by dividing 3500 by 50.Next, I'll calculate the number of Rs. 100 notes by subtracting the number of Rs. 50 notes from the total number of notes, which is 85.After finding the number of Rs. 100 notes, I'll calculate the total amount in these notes by multiplying the number of Rs. 100 notes by 100.Finally, I'll add the total amount in Rs. 100 notes to the total amount in Rs. 50 notes to find the overall total amount of all the currency notes.

💡To find the reciprocal of (-frac{1}{3}), I need to determine a number that, when multiplied by (-frac{1}{3}), equals 1.I start with the equation:[-frac{1}{3} times x = 1]To solve for (x), I multiply both sides by the reciprocal of (-frac{1}{3}), which is (-3):[-frac{1}{3} times (-3) = 1]This simplifies to:[1 = 1]Thus, the reciprocal of (-frac{1}{3}) is (-3).