💡Okay, so I've got this polynomial: 7x⁵ + 3e^{x²} + 4x³ + 15 + 8πx⁶ + 3√2x² + 2. I need to find its degree. Hmm, I remember that the degree of a polynomial is the highest power of x with a non-zero coefficient. But wait, I also recall that not all terms in an expression are necessarily polynomial terms. For example, exponential functions like e^{x²} aren't polynomials. So, I guess I should first identify which terms are actually part of the polynomial.Looking at the expression, the terms are:1. 7x⁵2. 3e^{x²}3. 4x³4. 155. 8πx⁶6. 3√2x²7. 2Now, 3e^{x²} is an exponential function, which isn't a polynomial term. So, I can disregard that. The constants 15 and 2 are also part of the polynomial, but they don't affect the degree since they're just x⁰ terms. So, focusing on the polynomial terms:- 7x⁵- 4x³- 8πx⁶- 3√2x²Now, I need to find the highest exponent here. Let's list the exponents:- 7x⁵ has an exponent of 5- 4x³ has an exponent of 3- 8πx⁶ has an exponent of 6- 3√2x² has an exponent of 2Comparing these exponents: 5, 3, 6, and 2. The highest one is 6. So, the term with the highest exponent is 8πx⁶, which means the degree of the polynomial is 6.Wait, just to make sure I didn't miss anything. The term 3e^{x²} was excluded because it's not a polynomial term, right? And constants don't affect the degree. So, yeah, 6 should be the correct degree.

💡First, I need to determine the total number of students in class (A) by considering the number of groups and the number of students in each group.There are 8 groups in total. Out of these, 6 groups have 6 students each, and the remaining 2 groups have 7 students each.To find the total number of students, I will calculate the number of students in the 6 groups with 6 students each and the number of students in the 2 groups with 7 students each, then add these two quantities together.Calculating the number of students in the 6 groups:6 groups × 6 students/group = 36 studentsCalculating the number of students in the 2 groups:2 groups × 7 students/group = 14 studentsAdding these together gives:36 students + 14 students = 50 studentsTherefore, the total number of students in class (A) is 50.

💡Okay, so I have this problem where I'm given that ( x = sqrt{7 - 4 sqrt{3}} ) and I need to find the value of ( a ) where ( frac{x^{2} - 4x + 5}{x^{2} - 4x + 3} = a ). Hmm, that seems a bit tricky, but let me try to break it down step by step.First, I know that ( x ) is given as a square root, so maybe I can simplify that expression. The expression inside the square root is ( 7 - 4 sqrt{3} ). I remember that sometimes expressions like this can be rewritten as a square of a binomial, like ( (sqrt{a} - sqrt{b})^2 ). Let me see if that works here.If I assume that ( 7 - 4 sqrt{3} = (sqrt{a} - sqrt{b})^2 ), then expanding the right side gives me ( a + b - 2sqrt{ab} ). Comparing this to ( 7 - 4 sqrt{3} ), I can set up two equations:1. ( a + b = 7 ) (the non-radical part)2. ( -2sqrt{ab} = -4sqrt{3} ) (the radical part)From the second equation, if I divide both sides by -2, I get ( sqrt{ab} = 2sqrt{3} ). Squaring both sides gives ( ab = 12 ).Now I have a system of equations:- ( a + b = 7 )- ( ab = 12 )This looks like a quadratic equation where ( a ) and ( b ) are the roots. The quadratic would be ( t^2 - (a + b)t + ab = 0 ), which becomes ( t^2 - 7t + 12 = 0 ). Let me solve this quadratic equation.Using the quadratic formula, ( t = frac{7 pm sqrt{49 - 48}}{2} = frac{7 pm 1}{2} ). So the solutions are ( t = 4 ) and ( t = 3 ). Therefore, ( a = 4 ) and ( b = 3 ), or vice versa.So, ( 7 - 4 sqrt{3} = (sqrt{4} - sqrt{3})^2 = (2 - sqrt{3})^2 ). That means ( x = sqrt{(2 - sqrt{3})^2} = 2 - sqrt{3} ). Okay, so ( x = 2 - sqrt{3} ).Now, I need to compute ( frac{x^{2} - 4x + 5}{x^{2} - 4x + 3} ). Let me first find ( x^2 ). Since ( x = 2 - sqrt{3} ), squaring it gives:( x^2 = (2 - sqrt{3})^2 = 4 - 4sqrt{3} + 3 = 7 - 4sqrt{3} ).So, ( x^2 = 7 - 4sqrt{3} ).Now, let me compute the numerator and denominator separately.Numerator: ( x^2 - 4x + 5 )Substituting ( x^2 = 7 - 4sqrt{3} ) and ( x = 2 - sqrt{3} ):( (7 - 4sqrt{3}) - 4(2 - sqrt{3}) + 5 )Let me compute each term:- ( 7 - 4sqrt{3} )- ( -4(2 - sqrt{3}) = -8 + 4sqrt{3} )- ( +5 )Adding them together:( 7 - 4sqrt{3} - 8 + 4sqrt{3} + 5 )Combine like terms:- Constants: ( 7 - 8 + 5 = 4 )- Radicals: ( -4sqrt{3} + 4sqrt{3} = 0 )So, the numerator simplifies to 4.Denominator: ( x^2 - 4x + 3 )Again, substituting ( x^2 = 7 - 4sqrt{3} ) and ( x = 2 - sqrt{3} ):( (7 - 4sqrt{3}) - 4(2 - sqrt{3}) + 3 )Compute each term:- ( 7 - 4sqrt{3} )- ( -4(2 - sqrt{3}) = -8 + 4sqrt{3} )- ( +3 )Adding them together:( 7 - 4sqrt{3} - 8 + 4sqrt{3} + 3 )Combine like terms:- Constants: ( 7 - 8 + 3 = 2 )- Radicals: ( -4sqrt{3} + 4sqrt{3} = 0 )So, the denominator simplifies to 2.Therefore, ( a = frac{4}{2} = 2 ).Wait, that seems straightforward, but let me double-check my steps to make sure I didn't make any mistakes.First, I simplified ( x = sqrt{7 - 4sqrt{3}} ) to ( 2 - sqrt{3} ). That seems correct because ( (2 - sqrt{3})^2 = 4 - 4sqrt{3} + 3 = 7 - 4sqrt{3} ).Then, I calculated ( x^2 = 7 - 4sqrt{3} ), which is consistent.For the numerator:( x^2 - 4x + 5 = (7 - 4sqrt{3}) - 4(2 - sqrt{3}) + 5 )Breaking it down:- ( 7 - 4sqrt{3} )- ( -8 + 4sqrt{3} )- ( +5 )Adding up: ( 7 - 8 + 5 = 4 ) and ( -4sqrt{3} + 4sqrt{3} = 0 ). So numerator is 4.Denominator:( x^2 - 4x + 3 = (7 - 4sqrt{3}) - 4(2 - sqrt{3}) + 3 )Breaking it down:- ( 7 - 4sqrt{3} )- ( -8 + 4sqrt{3} )- ( +3 )Adding up: ( 7 - 8 + 3 = 2 ) and ( -4sqrt{3} + 4sqrt{3} = 0 ). So denominator is 2.Thus, ( a = 4 / 2 = 2 ). That seems correct.Alternatively, maybe there's a smarter way without expanding everything. Let me think.I notice that both the numerator and the denominator have ( x^2 - 4x ). Maybe I can let ( y = x^2 - 4x ), then the expression becomes ( frac{y + 5}{y + 3} ).From earlier, I found that ( x = 2 - sqrt{3} ). Let me compute ( y = x^2 - 4x ).We already know ( x^2 = 7 - 4sqrt{3} ), so ( y = (7 - 4sqrt{3}) - 4(2 - sqrt{3}) ).Compute ( 4(2 - sqrt{3}) = 8 - 4sqrt{3} ).So, ( y = 7 - 4sqrt{3} - 8 + 4sqrt{3} = (7 - 8) + (-4sqrt{3} + 4sqrt{3}) = -1 + 0 = -1 ).Therefore, ( y = -1 ).So, the expression becomes ( frac{-1 + 5}{-1 + 3} = frac{4}{2} = 2 ). Yep, same result.That's a quicker way, using substitution. I like that method better because it avoids expanding all the terms.So, in summary, by recognizing that ( x ) can be expressed as ( 2 - sqrt{3} ), and then substituting into the given expression, I can simplify it to find that ( a = 2 ).I think that's solid. I don't see any mistakes in my reasoning, and both methods lead to the same answer. So I'm confident that ( a = 2 ).**Final Answer**The value of ( a ) is boxed{2}.

💡Okay, so I have this problem with two parts. Let me try to understand what it's asking and figure out how to approach it step by step.First, the problem mentions a point O at the origin of a rectangular coordinate system, and the positive semi-axis of the x-axis is the polar axis. Both coordinate systems have equal length units. There's a parametric equation for a line l and a polar equation for a curve C.Part (1) asks: If α = π/6, find the general equation of line l and the rectangular coordinate equation of curve C.Alright, so for part (1), I need to convert the parametric equation of line l into its general form when α is π/6. Then, I need to convert the polar equation of curve C into a rectangular (Cartesian) equation.Let me start with the parametric equation of line l:x = t cos αy = 2 + t sin αGiven α = π/6, so cos(π/6) is √3/2 and sin(π/6) is 1/2.So substituting these values in:x = t*(√3/2)y = 2 + t*(1/2)I need to eliminate the parameter t to find the general equation of the line.From the first equation, x = (√3/2) t, so t = (2x)/√3.Substitute this into the second equation:y = 2 + (1/2)*(2x)/√3Simplify:y = 2 + (x)/√3So that's the equation in slope-intercept form. To write it in general form, I can rearrange it:y - 2 = (x)/√3Multiply both sides by √3 to eliminate the denominator:√3(y - 2) = xWhich gives:x - √3 y + 2√3 = 0So that's the general equation of line l when α = π/6.Now, moving on to curve C. Its polar equation is given as ρ cos²θ = 4 sin θ.I need to convert this into a rectangular (Cartesian) equation.Recall that in polar coordinates:ρ cos θ = xρ sin θ = yAlso, ρ² = x² + y²Given the equation: ρ cos²θ = 4 sin θLet me rewrite this:ρ cos²θ = 4 sin θMultiply both sides by ρ:ρ² cos²θ = 4 ρ sin θNow, substitute the rectangular equivalents:ρ² cos²θ is (x² + y²) * (x² / (x² + y²))² ?Wait, no. Let me think again.Wait, cos θ = x / ρ, so cos²θ = x² / ρ².So, ρ² cos²θ = ρ² * (x² / ρ²) = x².Similarly, 4 ρ sin θ = 4 y.So, substituting back into the equation:x² = 4 ySo, the rectangular equation of curve C is x² = 4 y.That seems straightforward. So, curve C is a parabola opening upwards.Alright, so part (1) is done. The general equation of line l is x - √3 y + 2√3 = 0, and the rectangular equation of curve C is x² = 4 y.Now, moving on to part (2). It says: Suppose line l intersects curve C at points A and B. Find the minimum value of |AB| as α varies.So, we need to find the minimum distance between points A and B as the parameter α changes. That is, as the line l rotates around the point (0,2), since the parametric equation is x = t cos α, y = 2 + t sin α, which is a line passing through (0,2) with direction determined by α.So, the line l is a straight line passing through (0,2) with direction vector (cos α, sin α). So, as α varies, the line rotates around (0,2). We need to find the minimum distance between the two intersection points A and B of this line with the parabola x² = 4 y.To find |AB|, we can parametrize the line, substitute into the parabola equation, solve for the parameter t, find the two points A and B, compute the distance between them, and then find the minimum of this distance as α varies.Let me try to formalize this.Given the parametric equations:x = t cos αy = 2 + t sin αSubstitute these into the parabola equation x² = 4 y:(t cos α)^2 = 4 (2 + t sin α)Simplify:t² cos² α = 8 + 4 t sin αBring all terms to one side:t² cos² α - 4 t sin α - 8 = 0This is a quadratic equation in t:t² cos² α - 4 t sin α - 8 = 0Let me denote this as:A t² + B t + C = 0Where:A = cos² αB = -4 sin αC = -8So, the solutions for t are:t = [4 sin α ± sqrt( ( -4 sin α )² - 4 * cos² α * (-8) ) ] / (2 cos² α )Simplify discriminant D:D = (16 sin² α) + 32 cos² αFactor out 16:D = 16 (sin² α + 2 cos² α )Wait, 16 sin² α + 32 cos² α = 16 sin² α + 32 cos² αAlternatively, factor 16:= 16 (sin² α + 2 cos² α )But sin² α + cos² α = 1, so sin² α + 2 cos² α = 1 + cos² αTherefore, D = 16 (1 + cos² α )So, sqrt(D) = 4 sqrt(1 + cos² α )Thus, t = [4 sin α ± 4 sqrt(1 + cos² α ) ] / (2 cos² α )Simplify numerator and denominator:t = [4 (sin α ± sqrt(1 + cos² α )) ] / (2 cos² α )= 2 (sin α ± sqrt(1 + cos² α )) / cos² αTherefore, the two solutions for t are:t1 = 2 (sin α + sqrt(1 + cos² α )) / cos² αt2 = 2 (sin α - sqrt(1 + cos² α )) / cos² αSo, the two points A and B correspond to parameters t1 and t2.The distance between A and B is |AB|, which can be found by the distance between the two points on the line, which is |t1 - t2| times the direction vector's magnitude.But since the direction vector is (cos α, sin α), which has magnitude 1, because cos² α + sin² α = 1.Therefore, |AB| = |t1 - t2|Compute |t1 - t2|:t1 - t2 = [2 (sin α + sqrt(1 + cos² α )) / cos² α ] - [2 (sin α - sqrt(1 + cos² α )) / cos² α ]= [2 sin α + 2 sqrt(1 + cos² α ) - 2 sin α + 2 sqrt(1 + cos² α ) ] / cos² α= [4 sqrt(1 + cos² α ) ] / cos² αTherefore, |AB| = 4 sqrt(1 + cos² α ) / cos² αSo, |AB| = 4 sqrt(1 + cos² α ) / cos² αWe need to find the minimum value of this expression as α varies in [0, π). But we have to be careful because when α = π/2, cos α = 0, which would make the denominator zero, but in the original parametric equation, when α = π/2, the line is vertical, x = 0, and substituting into x² = 4 y, we get y = 0, so the line intersects the parabola only at (0,0). But since the line is vertical, it's a tangent to the parabola at (0,0). So, in this case, the line would intersect the parabola at only one point, so |AB| would be zero? Wait, but in our previous calculation, when α approaches π/2, cos α approaches zero, so |AB| tends to infinity. So, perhaps α cannot be π/2 because the line is tangent, so we have to consider α in [0, π/2) ∪ (π/2, π).So, to find the minimum of |AB| = 4 sqrt(1 + cos² α ) / cos² α for α ∈ [0, π/2) ∪ (π/2, π).Let me denote u = cos² α. Since α ∈ [0, π), cos² α ∈ (0,1]. Because when α = 0, cos² α = 1; when α approaches π/2, cos² α approaches 0; and when α approaches π, cos² α approaches 1 again.So, u ∈ (0,1].Express |AB| in terms of u:|AB| = 4 sqrt(1 + u) / uSo, we need to minimize f(u) = 4 sqrt(1 + u) / u for u ∈ (0,1].Let me compute the derivative of f(u) with respect to u to find the minimum.First, write f(u) as 4 (1 + u)^{1/2} / uCompute f'(u):f'(u) = 4 [ ( (1/2)(1 + u)^{-1/2} * 1 ) * u - (1 + u)^{1/2} * 1 ] / u²Simplify:= 4 [ ( (1)/(2 sqrt(1 + u)) ) * u - sqrt(1 + u) ] / u²= 4 [ ( u / (2 sqrt(1 + u)) ) - sqrt(1 + u) ] / u²Factor out sqrt(1 + u) in the numerator:= 4 [ ( u / (2 sqrt(1 + u)) ) - sqrt(1 + u) ] / u²= 4 [ ( u - 2(1 + u) ) / (2 sqrt(1 + u)) ] / u²Simplify numerator:u - 2(1 + u) = u - 2 - 2u = -u - 2So,f'(u) = 4 [ (-u - 2) / (2 sqrt(1 + u)) ] / u²= 4 * (-u - 2) / (2 sqrt(1 + u) u² )= (-2(u + 2)) / ( sqrt(1 + u) u² )So, f'(u) = -2(u + 2) / ( sqrt(1 + u) u² )Since u ∈ (0,1], the denominator sqrt(1 + u) u² is always positive. The numerator is -2(u + 2), which is always negative because u + 2 > 0. Therefore, f'(u) is always negative in the interval u ∈ (0,1].This means that f(u) is decreasing on (0,1]. Therefore, the minimum occurs at the right endpoint, which is u = 1.So, the minimum value of |AB| is f(1) = 4 sqrt(1 + 1) / 1 = 4 sqrt(2).Therefore, the minimum value of |AB| is 4√2.Wait, but let me verify this because sometimes endpoints can be tricky. When u approaches 0, f(u) approaches infinity, and at u =1, f(u) is 4√2. Since f(u) is decreasing, yes, the minimum is at u=1.So, when u=1, cos² α =1, which implies cos α = ±1, but since α ∈ [0, π), cos α =1 when α=0.So, when α=0, the line l is horizontal, y=2, because the parametric equations become x = t, y=2 + t*0=2.So, the line y=2 intersects the parabola x²=4 y at points where x²=8, so x=±2√2, y=2.Thus, points A and B are (-2√2, 2) and (2√2, 2), so the distance |AB| is 4√2, which matches our calculation.Therefore, the minimum value of |AB| is indeed 4√2.So, summarizing:(1) When α=π/6, the general equation of line l is x - √3 y + 2√3 =0, and the rectangular equation of curve C is x²=4 y.(2) The minimum value of |AB| as α varies is 4√2.**Final Answer**The minimum value of |AB| is boxed{4sqrt{2}}.

💡Okay, so I need to find the 150th term of a sequence where we remove all perfect squares and multiples of 5 from the positive integers. Hmm, that sounds a bit tricky, but let me break it down step by step.First, I should understand what the sequence looks like. Normally, positive integers go 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on. But in this case, we're omitting two types of numbers: perfect squares and multiples of 5. So, perfect squares are numbers like 1, 4, 9, 16, 25, etc., and multiples of 5 are 5, 10, 15, 20, 25, 30, etc.Wait a minute, I notice that some numbers are both perfect squares and multiples of 5, like 25, 100, 225, etc. So, if I just subtract the count of perfect squares and multiples of 5 separately, I might be double-counting those overlaps. I need to account for that to avoid overcounting.Alright, so let's figure out how many numbers we're omitting up to a certain point. Let's say I want to find the 150th term in this modified sequence. That means I need to find the position in the original sequence of integers that corresponds to the 150th term after removing the perfect squares and multiples of 5.To do this, I need to calculate how many numbers are omitted up to a certain number N. Then, I can set up an equation where N minus the number of omitted numbers equals 150. Solving for N will give me the 150th term.Let me formalize this. Let N be the number in the original sequence that corresponds to the 150th term in the modified sequence. The number of perfect squares up to N is the floor of the square root of N, which is denoted as ⎣√N⎦. The number of multiples of 5 up to N is the floor of N divided by 5, denoted as ⎣N/5⎦. However, as I thought earlier, some numbers are both perfect squares and multiples of 5, so I need to subtract those overlaps to avoid double-counting. The number of such overlaps is the number of perfect squares that are multiples of 5, which are the squares of multiples of 5. These are 25, 100, 225, etc., so their count up to N is the floor of the square root of N divided by 5, denoted as ⎣√(N)/5⎦.Putting this all together, the total number of omitted numbers up to N is:Number of perfect squares + Number of multiples of 5 - Number of overlapsWhich is:⎣√N⎦ + ⎣N/5⎦ - ⎣√(N)/5⎦So, the equation I need to solve is:N - (⎣√N⎦ + ⎣N/5⎦ - ⎣√(N)/5⎦) = 150This equation looks a bit complicated, but maybe I can estimate N first and then adjust.Let's make an initial guess. If there were no omissions, the 150th term would be 150. But since we're omitting some numbers, N must be larger than 150. Let's say N is around 150 plus the number of omissions up to 150.Calculating the number of omissions up to 150:Number of perfect squares up to 150: ⎣√150⎦ = 12 (since 12^2 = 144 and 13^2 = 169)Number of multiples of 5 up to 150: ⎣150/5⎦ = 30Number of overlaps (perfect squares that are multiples of 5 up to 150): ⎣√150/5⎦ = ⎣12.247/5⎦ = ⎣2.449⎦ = 2 (since 25 and 100 are the overlaps)So, total omissions up to 150: 12 + 30 - 2 = 40Therefore, N is approximately 150 + 40 = 190. Let's check if 190 is the correct N.Calculating the number of omissions up to 190:Number of perfect squares up to 190: ⎣√190⎦ = 13 (since 13^2 = 169 and 14^2 = 196)Number of multiples of 5 up to 190: ⎣190/5⎦ = 38Number of overlaps (perfect squares that are multiples of 5 up to 190): ⎣√190/5⎦ = ⎣13.784/5⎦ = ⎣2.756⎦ = 2 (still 25 and 100, since 225 is beyond 190)Total omissions up to 190: 13 + 38 - 2 = 49So, the number of terms in the modified sequence up to 190 is 190 - 49 = 141. But we need the 150th term, so 141 is still less than 150. Therefore, N needs to be larger than 190.Let's try N = 200.Calculating the number of omissions up to 200:Number of perfect squares up to 200: ⎣√200⎦ = 14 (since 14^2 = 196 and 15^2 = 225)Number of multiples of 5 up to 200: ⎣200/5⎦ = 40Number of overlaps (perfect squares that are multiples of 5 up to 200): ⎣√200/5⎦ = ⎣14.142/5⎦ = ⎣2.828⎦ = 2 (still 25 and 100, 225 is beyond 200)Total omissions up to 200: 14 + 40 - 2 = 52Number of terms in the modified sequence up to 200: 200 - 52 = 148. Still less than 150. So, N needs to be a bit larger.Let's try N = 205.Calculating the number of omissions up to 205:Number of perfect squares up to 205: ⎣√205⎦ = 14 (since 14^2 = 196 and 15^2 = 225)Number of multiples of 5 up to 205: ⎣205/5⎦ = 41Number of overlaps (perfect squares that are multiples of 5 up to 205): ⎣√205/5⎦ = ⎣14.318/5⎦ = ⎣2.863⎦ = 2Total omissions up to 205: 14 + 41 - 2 = 53Number of terms in the modified sequence up to 205: 205 - 53 = 152. Now, 152 is more than 150, so the 150th term is somewhere between 190 and 205.Let's try N = 200, which gave us 148 terms. So, from 200 to 205, we have 5 numbers: 201, 202, 203, 204, 205.We need 2 more terms to reach 150 from 148. Let's check each number:201: Is 201 a perfect square? No. Is it a multiple of 5? No. So, it's included. That's 149th term.202: Similarly, not a perfect square or multiple of 5. Included. That's 150th term.Wait, so 202 is the 150th term? Let me double-check.Wait, when N=200, we have 148 terms. Then, 201 is 149th, and 202 is 150th. So, yes, 202 is the 150th term.But earlier, I thought N=190 gives 141 terms, and N=200 gives 148, so the difference is 7 terms between 190 and 200. Then, from 200 to 202, we add 2 more terms to reach 150.Wait, but let me verify the counts again because sometimes overlaps can affect the count.Wait, when N=200, the number of perfect squares is 14, multiples of 5 is 40, overlaps are 2, so total omissions are 52. So, 200 - 52 = 148.Then, from 201 to 205, we have 5 numbers. Let's see which of these are omitted:201: Not a perfect square, not a multiple of 5. Included.202: Same as above. Included.203: Same. Included.204: Same. Included.205: Is 205 a multiple of 5? Yes, 205 ÷ 5 = 41. So, 205 is omitted.So, from 201 to 205, only 205 is omitted. So, 4 numbers are included: 201, 202, 203, 204.Therefore, adding 4 to 148 gives 152, which matches the earlier count.So, the 150th term is 202.Wait, but earlier I thought N=205 gives 152 terms, so 202 is the 150th term.But let me check another way. Maybe I can use the inclusion-exclusion principle more carefully.Total numbers up to N: NMinus perfect squares: ⎣√N⎦Minus multiples of 5: ⎣N/5⎦Plus overlaps (since they were subtracted twice): ⎣√(N)/5⎦So, total included numbers: N - ⎣√N⎦ - ⎣N/5⎦ + ⎣√(N)/5⎦We need this to be equal to 150.So, N - ⎣√N⎦ - ⎣N/5⎦ + ⎣√(N)/5⎦ = 150Let me try N=202.Calculate:⎣√202⎦ = 14 (since 14^2=196, 15^2=225)⎣202/5⎦ = 40 (since 202 ÷ 5 = 40.4)⎣√202/5⎦ = ⎣14.212/5⎦ = ⎣2.842⎦ = 2So, total included numbers: 202 - 14 - 40 + 2 = 202 - 54 + 2 = 202 - 52 = 150Yes, that works. So, N=202 is the 150th term.Wait, so earlier I thought N=202 is the 150th term, and the calculation confirms it.But earlier, when I tried N=200, I got 148 terms, and then adding 201 and 202 gives 150. So, 202 is indeed the 150th term.But wait, in my initial estimation, I thought N=190 gives 141 terms, and N=200 gives 148, so the difference is 7 terms over 10 numbers, which seems consistent.So, yes, the 150th term is 202.But let me double-check by listing some numbers around 200 to ensure.Let's list numbers from 190 to 205 and see which are included:190: multiple of 5? Yes, 190 ÷ 5 = 38. So, omitted.191: Not a perfect square or multiple of 5. Included.192: Same. Included.193: Same. Included.194: Same. Included.195: Multiple of 5. Omitted.196: Perfect square (14^2). Omitted.197: Included.198: Included.199: Included.200: Multiple of 5. Omitted.201: Included.202: Included.203: Included.204: Included.205: Multiple of 5. Omitted.So, from 190 to 205, the included numbers are:191, 192, 193, 194, 197, 198, 199, 201, 202, 203, 204.That's 11 numbers. But wait, from 190 to 205, there are 16 numbers. Out of these, 190, 195, 196, 200, 205 are omitted. So, 5 omissions, leaving 11 included numbers.But earlier, when N=200, we had 148 terms, and N=205 gives 152. So, from 200 to 205, we add 5 numbers, but only 4 are included (201, 202, 203, 204), since 205 is omitted.Wait, but in the list above, from 190 to 205, we have 11 included numbers, but from 200 to 205, only 4 are included. So, from 190 to 200, we have 11 - 4 = 7 included numbers. That matches the earlier count where N=200 gives 148 terms, which is 141 (up to 190) plus 7 more.So, yes, the 150th term is 202.But wait, let me check another way. Let's use the formula:Included numbers up to N = N - ⎣√N⎦ - ⎣N/5⎦ + ⎣√(N)/5⎦We need this to be 150.Let me try N=202:Included = 202 - 14 - 40 + 2 = 202 - 54 + 2 = 150. Perfect.If I try N=201:Included = 201 - 14 - 40 + 2 = 201 - 54 + 2 = 149. So, 201 is the 149th term.Therefore, 202 is indeed the 150th term.I think that's solid. So, the answer is 202.

💡Okay, so I have this problem with two people, A and B, each shooting at a target. A has a probability of 2/3 to hit, and B has a probability of 3/4 to hit. The shots are independent, meaning one person's shot doesn't affect the other's. There are three parts to this problem, and I need to solve each one step by step.Starting with part (1): Find the probability that both A and B hit the target with one shot each.Hmm, since both events are independent, I think I can just multiply their probabilities. So, the probability that A hits is 2/3, and the probability that B hits is 3/4. So, multiplying them together should give me the probability that both hit. Let me write that down:P(A and B) = P(A) * P(B) = (2/3) * (3/4) = ?Calculating that, 2/3 times 3/4. The 3s cancel out, so it's 2/4, which simplifies to 1/2. So, the probability that both hit is 1/2. That seems straightforward.Moving on to part (2): Find the probability that A shoots 4 times and hits the target exactly 3 times in a row.Wait, exactly 3 times in a row? So, does that mean 3 consecutive hits and then a miss? Or could it be any 3 hits in a row within the 4 shots?I think it means that A has exactly 3 consecutive hits somewhere in the 4 shots. So, there are two possibilities:1. A hits the first three shots and misses the fourth.2. A misses the first shot and then hits the next three.These are the two scenarios where there are exactly 3 consecutive hits in 4 shots. Since these are mutually exclusive events, I can calculate the probability for each and then add them together.Let me denote the probability of hitting as P(H) = 2/3 and missing as P(M) = 1 - 2/3 = 1/3.First scenario: H, H, H, M.Probability = (2/3) * (2/3) * (2/3) * (1/3) = (2/3)^3 * (1/3) = 8/81 * 1/3 = 8/243? Wait, no, that's not right. Wait, (2/3)^3 is 8/27, multiplied by 1/3 is 8/81.Second scenario: M, H, H, H.Probability = (1/3) * (2/3) * (2/3) * (2/3) = (1/3) * (8/27) = 8/81.So, adding both scenarios together: 8/81 + 8/81 = 16/81.So, the probability is 16/81. That seems correct.Now, part (3): If B will be terminated after missing the target consecutively 2 times during shooting, find the probability that B will be terminated after shooting exactly 4 times.Alright, so B is shooting until he misses twice in a row, and we need the probability that this happens exactly on the 4th shot. That means that in the first 3 shots, B hasn't been terminated yet, and then on the 4th shot, he gets the second consecutive miss.Let me think about the possible sequences of shots that would lead to termination on the 4th shot.First, B hasn't been terminated before the 4th shot, so in the first 3 shots, he hasn't had two consecutive misses. Then, on the 4th shot, he gets a miss, which would make it two consecutive misses if the 3rd shot was also a miss.Wait, so the 4th shot is the second miss, so the 3rd shot must have been a miss as well. But before that, in the first two shots, he couldn't have two consecutive misses.So, the possible sequences are:1. Hit, Hit, Miss, Miss2. Hit, Miss, Hit, Miss, Miss? Wait, no, because we're only considering up to 4 shots.Wait, no, let's think again. To terminate on the 4th shot, the 4th shot must be a miss, and the 3rd shot must be a miss as well, making it two consecutive misses. But before that, in the first two shots, he couldn't have two consecutive misses.So, the first two shots can be:- Both hits: H, H- One hit and one miss, but not two misses in a row.So, the possible sequences are:1. H, H, M, M2. H, M, H, M, M? Wait, no, that's five shots. We need exactly four shots.Wait, no, let's think about it. To terminate on the 4th shot, the 3rd and 4th shots must be misses. So, the first two shots can be any combination except two consecutive misses.So, the first two shots can be:- H, H- H, M- M, HBut not M, M because that would terminate him on the second shot.So, for each of these possibilities, followed by M, M on the third and fourth shots.So, the possible sequences are:1. H, H, M, M2. H, M, M, M3. M, H, M, MWait, but wait, in the second case, H, M, M, M: after the second shot, he has a miss, then another miss on the third shot, which would terminate him on the third shot, right? Because two consecutive misses.But we need him to be terminated on the fourth shot, so the two consecutive misses must occur at the third and fourth shots. So, the third shot must be a miss, and the fourth shot must be a miss, but the second shot cannot be a miss if the third is a miss because that would have terminated him earlier.Wait, no. Let me clarify.If he misses on the second and third shots, he would be terminated on the third shot. So, to be terminated on the fourth shot, the two consecutive misses must be on the third and fourth shots, but the second shot must not be a miss if the third is a miss.Wait, that might not make sense. Let me think again.To terminate on the fourth shot, the last two shots must be misses, and before that, he hasn't had two consecutive misses.So, the first two shots can be anything except two consecutive misses, and then the third shot is a miss, and the fourth shot is a miss.So, the possible sequences are:1. H, H, M, M2. H, M, M, M: but wait, in this case, the second and third shots are misses, which would terminate him on the third shot, so this sequence is invalid because he would have been terminated earlier.3. M, H, M, M: here, the first shot is a miss, second is a hit, third is a miss, fourth is a miss. So, the third and fourth are misses, but the first and third are misses, but not consecutive. So, this is a valid sequence.Wait, but in this case, the first shot is a miss, then a hit, then two misses. So, the two consecutive misses are at the third and fourth shots, and before that, he didn't have two consecutive misses. So, this is valid.Similarly, the sequence H, H, M, M is valid because the first two are hits, then two misses.But the sequence H, M, M, M is invalid because he would have been terminated on the third shot.So, only two valid sequences:1. H, H, M, M2. M, H, M, MWait, but what about the sequence M, M, H, M? No, because if he misses the first two shots, he's terminated on the second shot.Similarly, H, M, H, M, M is five shots, which is beyond our consideration.So, only two valid sequences: H, H, M, M and M, H, M, M.So, let's calculate the probabilities for these two sequences.First sequence: H, H, M, MProbability = P(H) * P(H) * P(M) * P(M) = (3/4) * (3/4) * (1/4) * (1/4) = (9/16) * (1/16) = 9/256.Wait, no, that's not correct. Wait, (3/4)^2 * (1/4)^2 = 9/16 * 1/16 = 9/256.Second sequence: M, H, M, MProbability = P(M) * P(H) * P(M) * P(M) = (1/4) * (3/4) * (1/4) * (1/4) = (3/4) * (1/4)^3 = 3/256.Wait, no, let's calculate it step by step:(1/4) * (3/4) * (1/4) * (1/4) = (1/4) * (3/4) * (1/4) * (1/4) = (3) / (4^4) = 3/256.So, total probability is 9/256 + 3/256 = 12/256 = 3/64.So, the probability that B is terminated after exactly 4 shots is 3/64.Wait, let me double-check.First sequence: H, H, M, M: (3/4)^2 * (1/4)^2 = 9/256.Second sequence: M, H, M, M: (1/4) * (3/4) * (1/4)^2 = 3/256.Adding them: 9 + 3 = 12, so 12/256 = 3/64. Yes, that seems correct.So, summarizing:1. Both A and B hit: 1/2.2. A hits exactly 3 times in a row in 4 shots: 16/81.3. B is terminated after exactly 4 shots: 3/64.I think that's all.