answer：To estimate the total number of fish in the pond, I can use the Lincoln-Petersen index method.First, I'll note the number of fish initially tagged and released back into the pond, which is 100.Next, I'll observe the number of fish caught in the second sample, which is 300, and determine how many of those were tagged, which is 15.Using the Lincoln-Petersen formula, the estimated total population (N) can be calculated by multiplying the number of tagged fish in the first sample (M) by the total number of fish caught in the second sample (C), and then dividing by the number of tagged fish recaptured in the second sample (R).So, N = (M × C) / R.Plugging in the numbers: N = (100 × 300) / 15.Calculating this gives N = 2000.Therefore, the estimated number of fish in the pond is 2000.

answer：Okay, so I have this problem here: three lines, y equals 2x, x plus y equals 3, and mx minus 2y minus 5 equals 0. They all intersect at the same point, and I need to find the value of m. Hmm, let me think about how to approach this.First, if all three lines intersect at the same point, that means there's a single point (x, y) that satisfies all three equations. So, if I can find the point where the first two lines intersect, that should also lie on the third line. Then, I can plug that point into the third equation to solve for m. That makes sense.Alright, let's start by finding the intersection point of the first two lines: y = 2x and x + y = 3. Since y is given in terms of x in the first equation, I can substitute y = 2x into the second equation. Let me write that down.Substituting y = 2x into x + y = 3 gives:x + (2x) = 3Simplify that:3x = 3Divide both sides by 3:x = 1Okay, so x is 1. Now, plug that back into y = 2x to find y:y = 2(1) = 2So, the intersection point of the first two lines is (1, 2). That means this point should also lie on the third line, which is mx - 2y - 5 = 0. Let me substitute x = 1 and y = 2 into that equation.Substituting into mx - 2y - 5 = 0:m(1) - 2(2) - 5 = 0Simplify:m - 4 - 5 = 0Combine like terms:m - 9 = 0Add 9 to both sides:m = 9Wait, so m is 9? Let me double-check that to make sure I didn't make a mistake.Starting again, the intersection point is (1, 2). Plugging into the third equation:m*1 - 2*2 - 5 = 0So, m - 4 - 5 = 0Which simplifies to m - 9 = 0Thus, m = 9. Yeah, that seems right.Just to be thorough, let me verify that all three lines actually intersect at (1, 2).First line: y = 2x. Plugging in x = 1, y = 2. Yep, that works.Second line: x + y = 3. 1 + 2 = 3. That's correct.Third line: mx - 2y - 5 = 0. With m = 9, x = 1, y = 2:9*1 - 2*2 - 5 = 9 - 4 - 5 = 0. Perfect, that works too.So, everything checks out. I think m is definitely 9.I guess another way to think about it is solving the system of equations. Since all three lines intersect at the same point, solving any two equations should give the intersection point, which we can then use to find m. That's exactly what I did.Alternatively, if I didn't substitute, I could solve the system using another method, like elimination or matrices, but substitution seems straightforward here because the first equation is already solved for y.Let me try solving the first two equations another way just for practice. The first equation is y = 2x, and the second is x + y = 3. If I write them together:1) y = 2x2) x + y = 3From equation 1, y is 2x, so equation 2 becomes x + 2x = 3, which is 3x = 3, so x = 1, same as before. Then y = 2(1) = 2. So, same result.Alternatively, if I didn't use substitution, I could have used elimination. Let's see:From equation 1: y = 2x. Let's write equation 2 as x + y = 3. If I subtract equation 1 from equation 2, I get:x + y - (y) = 3 - 2xSimplify:x = 3 - 2xAdd 2x to both sides:3x = 3x = 1Same result. So, regardless of the method, I end up with x = 1, y = 2.Therefore, plugging into the third equation, m must be 9. I think that's solid.Just to recap:1. Find intersection of first two lines: (1, 2)2. Plug into third line equation: m*1 - 2*2 - 5 = 03. Solve for m: m = 9Yep, that's the process. I don't see any mistakes in my reasoning, so I feel confident that m is 9.**Final Answer**The value of ( m ) is boxed{9}.

answer：Okay, so I have this problem here: I need to find the minimum positive period of the function ( y = 2 cos^2 x + 1 ). The options are A. ( frac{pi}{2} ), B. ( pi ), C. ( 2pi ), and D. ( 4pi ). Hmm, let me think about how to approach this.First, I remember that the period of a function is the smallest positive number ( T ) such that ( f(x + T) = f(x) ) for all ( x ). So, for trigonometric functions like cosine, the standard period is ( 2pi ). But in this case, the function is ( 2 cos^2 x + 1 ), which is a bit more complicated. Maybe I can simplify it using a trigonometric identity.I recall that ( cos^2 x ) can be expressed using the double-angle identity. Let me write that down: ( cos^2 x = frac{1 + cos 2x}{2} ). So, substituting that into the original function, we get:( y = 2 left( frac{1 + cos 2x}{2} right) + 1 ).Simplifying this, the 2 and the denominator 2 cancel out, so:( y = (1 + cos 2x) + 1 ).Adding the constants together:( y = 2 + cos 2x ).Okay, so now the function is simplified to ( y = 2 + cos 2x ). That looks simpler, and I can analyze its period more easily.I know that the period of ( cos kx ) is ( frac{2pi}{k} ). In this case, ( k = 2 ), so the period of ( cos 2x ) is ( frac{2pi}{2} = pi ). Since the function ( y = 2 + cos 2x ) is just a vertical shift of ( cos 2x ), the period doesn't change. So, the period of this function should be ( pi ).Wait, let me double-check that. The original function was ( 2 cos^2 x + 1 ), and after simplifying, it's ( 2 + cos 2x ). So, the period is indeed determined by the ( cos 2x ) term, which has a period of ( pi ). Therefore, the minimum positive period of the function is ( pi ).But to make sure I'm not making a mistake, let me test it with some values. Let's pick ( x = 0 ). Then, ( y = 2 cos^2 0 + 1 = 2(1) + 1 = 3 ). Now, if I add ( pi ) to ( x ), so ( x = pi ), then ( y = 2 cos^2 pi + 1 = 2(-1)^2 + 1 = 2(1) + 1 = 3 ). So, the function value is the same at ( x = 0 ) and ( x = pi ).What about halfway, say ( x = frac{pi}{2} )? Then, ( y = 2 cos^2 frac{pi}{2} + 1 = 2(0)^2 + 1 = 1 ). Adding ( pi ) to ( x ), so ( x = frac{3pi}{2} ), then ( y = 2 cos^2 frac{3pi}{2} + 1 = 2(0)^2 + 1 = 1 ). Again, the function value is the same.Let me check another point, like ( x = frac{pi}{4} ). Then, ( y = 2 cos^2 frac{pi}{4} + 1 = 2 left( frac{sqrt{2}}{2} right)^2 + 1 = 2 left( frac{1}{2} right) + 1 = 1 + 1 = 2 ). Adding ( pi ), so ( x = frac{5pi}{4} ), ( y = 2 cos^2 frac{5pi}{4} + 1 = 2 left( -frac{sqrt{2}}{2} right)^2 + 1 = 2 left( frac{1}{2} right) + 1 = 2 ). Same value again.Hmm, so it seems consistent. The function repeats its values every ( pi ) units. But just to be thorough, let me see if there's a smaller period. The options include ( frac{pi}{2} ). Let's test that.Take ( x = 0 ), ( y = 3 ). Then, ( x = frac{pi}{2} ), ( y = 1 ). If the period were ( frac{pi}{2} ), then ( fleft( frac{pi}{2} right) ) should equal ( f(0) ), but 1 ≠ 3. So, ( frac{pi}{2} ) can't be the period.What about ( frac{pi}{4} )? Let's see. ( x = 0 ), ( y = 3 ). ( x = frac{pi}{4} ), ( y = 2 ). ( x = frac{pi}{2} ), ( y = 1 ). ( x = frac{3pi}{4} ), ( y = 2 ). ( x = pi ), ( y = 3 ). So, it seems like the function goes from 3 to 2 to 1 to 2 to 3 over an interval of ( pi ). So, the period isn't smaller than ( pi ), because the function doesn't repeat its values before that.Therefore, I can be confident that the minimum positive period is ( pi ), which is option B.**Final Answer**boxed{B}

answer：Okay, so I have this problem here where I'm told that the solution to a system of linear equations is x=2 and y=1. The system is given as:ax + by = 7 ax - by = 1And I need to find the value of a - b. The options are A: 3, B: 2, C: 1, D: -1.Hmm, let me think about how to approach this. Since x=2 and y=1 is a solution, that means when I plug these values into both equations, they should satisfy them. So, I can substitute x=2 and y=1 into both equations and get two equations in terms of a and b. Then, I can solve those equations to find the values of a and b, and then compute a - b.Alright, let's write down the equations after substitution.First equation: a*(2) + b*(1) = 7 That simplifies to 2a + b = 7.Second equation: a*(2) - b*(1) = 1 That simplifies to 2a - b = 1.So now, I have the system:2a + b = 7 2a - b = 1Hmm, okay, so two equations with two variables. I can solve this system using either substitution or elimination. Let's see, elimination might be straightforward here because the coefficients of b are +1 and -1, so if I add the two equations, the b terms will cancel out.Let me add the two equations:(2a + b) + (2a - b) = 7 + 1 Simplify the left side: 2a + b + 2a - b = 4a Right side: 8So, 4a = 8 Divide both sides by 4: a = 2.Okay, so a is 2. Now, I can substitute a back into one of the original equations to find b. Let's take the first equation: 2a + b = 7.Substituting a=2: 2*(2) + b = 7 Which is 4 + b = 7 Subtract 4 from both sides: b = 3.So, a is 2 and b is 3. Therefore, a - b is 2 - 3, which is -1.Wait, but let me double-check my calculations to make sure I didn't make a mistake. It's easy to mix up signs or coefficients.Starting again, plugging x=2 and y=1 into the first equation:2a + b = 7 Second equation: 2a - b = 1.Adding them: 4a = 8, so a=2. Then, plugging back into first equation: 4 + b = 7, so b=3. Therefore, a - b is 2 - 3 = -1. So, the answer should be D: -1.But wait, let me also check if these values satisfy the second equation. If a=2 and b=3, then the second equation is 2*(2) - 3 = 4 - 3 = 1, which matches the given equation. So, that's correct.Alternatively, maybe I can solve for b first. Let's subtract the second equation from the first.First equation: 2a + b = 7 Second equation: 2a - b = 1 Subtracting the second from the first: (2a + b) - (2a - b) = 7 - 1 Simplify: 2a + b - 2a + b = 6 Which is 2b = 6 So, b = 3.Then, plugging back into one of the equations: 2a + 3 = 7 So, 2a = 4 a = 2.Same result. So, a=2, b=3, so a - b = -1.Therefore, the correct answer is D: -1.**Final Answer**boxed{D}

answer：Okay, so I have this problem where I need to use a thin wire to enclose a rectangle with an area of 4. There are two parts: first, expressing the total length of the wire as a function of a side length x, and second, analyzing that function to find where it's minimized.Starting with part 1: I need to express the total length of the wire, which is the perimeter of the rectangle, as a function of x. Let me recall that the perimeter of a rectangle is given by P = 2*(length + width). Since it's a rectangle, opposite sides are equal. Let's say one side is x, then the other side would be something else. But the area is given as 4, so area = length * width = x * y = 4, where y is the other side.So, if I let x be one side, the other side y can be expressed as 4/x. That makes sense because if x increases, y decreases to keep the area constant at 4.Now, the perimeter is 2*(x + y). Substituting y with 4/x, the perimeter becomes 2*(x + 4/x). So, the total length of the wire, which is the perimeter, is y = 2x + 8/x. Wait, hold on, that seems a bit off. Let me double-check.Wait, no, 2*(x + 4/x) is equal to 2x + 8/x. Hmm, actually, that's correct. So, the function y(x) is 2x + 8/x. But wait, in the second part of the problem, they mention f(x) = x + 4/x. That seems different. Maybe I made a mistake.Wait, let me think again. The perimeter is 2*(x + y). Since the area is 4, y = 4/x. So, substituting, perimeter is 2*(x + 4/x) = 2x + 8/x. So, that's correct. But in part 2, they refer to f(x) = x + 4/x. Hmm, maybe they simplified it by dividing by 2 or something? Or perhaps I misread the problem.Wait, no, the problem says "the total length of the wire, y, as a function of a side length x." So, I think my expression is correct: y = 2x + 8/x. But in part 2, they talk about f(x) = x + 4/x. Maybe they just scaled it down? Or maybe I need to adjust my function.Wait, perhaps I misread the problem. Let me check again. It says "enclose a rectangle with an area of 4." So, area is 4, so x*y = 4, so y = 4/x. Then, perimeter is 2x + 2y = 2x + 8/x. So, that's correct. But in part 2, they mention f(x) = x + 4/x. Maybe they are considering half the perimeter or something? Or perhaps I need to adjust my function.Wait, maybe I made a mistake in the first part. Let me think again. If the area is 4, and one side is x, the other side is 4/x. Then, the perimeter is 2*(x + 4/x). So, that's 2x + 8/x. So, the function is y = 2x + 8/x. But in part 2, they refer to f(x) = x + 4/x. Maybe they are considering half the perimeter? Or perhaps I need to adjust.Wait, maybe I need to express y as a function of x, but perhaps they consider the semi-perimeter or something else. Alternatively, maybe they made a typo or something. But in any case, let's proceed.So, for part 1, I think the correct expression is y = 2x + 8/x.But in part 2, they mention f(x) = x + 4/x. So, perhaps they are considering half of the perimeter? Because 2x + 8/x divided by 2 is x + 4/x. So, maybe they are considering half the perimeter? Or perhaps I need to adjust my function.Wait, maybe I misread the problem. Let me check again: "Use a thin wire to enclose a rectangle with an area of 4." So, the wire is the perimeter. So, the total length of the wire is the perimeter. So, I think my initial function is correct: y = 2x + 8/x.But in part 2, they refer to f(x) = x + 4/x. So, perhaps they are considering half the perimeter? Or maybe I need to adjust my function.Wait, maybe I need to express y as a function of x, but perhaps they are considering the perimeter as 2*(x + y), and since y = 4/x, then perimeter is 2x + 8/x, which is correct. So, perhaps in part 2, they are referring to a different function, maybe half the perimeter? Or perhaps I made a mistake.Wait, perhaps I need to express y as a function of x, but perhaps they are considering the perimeter as 2*(x + y), so y = 2x + 8/x. So, that's correct. So, maybe in part 2, they are referring to a different function, but perhaps I need to proceed with my function.Wait, but in part 2, they mention f(x) = x + 4/x, so maybe I need to adjust my function accordingly. Let me think.Wait, perhaps I need to express the perimeter in terms of x, but maybe they are considering the perimeter as 2*(x + y) = 2x + 2y, and since y = 4/x, then perimeter is 2x + 8/x. So, that's correct. So, perhaps in part 2, they are referring to a different function, but I think my function is correct.Wait, perhaps they made a mistake in the problem statement? Or maybe I need to adjust.Wait, no, maybe I need to consider that the function f(x) = x + 4/x is half the perimeter. Because if I take half the perimeter, it's x + 4/x. So, perhaps in part 2, they are referring to half the perimeter, and then I can use that to find the minimum.But in any case, let's proceed. So, for part 1, I think the correct function is y = 2x + 8/x.But in part 2, they refer to f(x) = x + 4/x, so perhaps they are considering half the perimeter? Or maybe I need to adjust.Wait, perhaps I need to express y as a function of x, but perhaps they are considering the perimeter as 2*(x + y), so y = 2x + 8/x, which is correct. So, perhaps in part 2, they are referring to a different function, but I think my function is correct.Wait, perhaps I need to proceed with my function, y = 2x + 8/x, and then in part 2, analyze that function.But in part 2, they mention f(x) = x + 4/x, so perhaps I need to adjust my function accordingly.Wait, maybe I made a mistake in part 1. Let me think again.If the area is 4, then x*y = 4, so y = 4/x. Then, the perimeter is 2*(x + y) = 2x + 2*(4/x) = 2x + 8/x. So, that's correct. So, y = 2x + 8/x.But in part 2, they refer to f(x) = x + 4/x. So, perhaps they are considering half the perimeter? Because 2x + 8/x divided by 2 is x + 4/x. So, maybe in part 2, they are referring to half the perimeter, which is x + 4/x, and then analyzing that function.But in any case, perhaps I can proceed with my function, y = 2x + 8/x, and then in part 2, analyze that function.Alternatively, perhaps I need to adjust my function to match what they have in part 2. Let me think.Wait, perhaps I need to express y as a function of x, but perhaps they are considering the perimeter as 2*(x + y) = 2x + 2y, and since y = 4/x, then perimeter is 2x + 8/x. So, that's correct. So, perhaps in part 2, they are referring to a different function, but I think my function is correct.Wait, perhaps I need to consider that the function f(x) = x + 4/x is half the perimeter, so if I can analyze that function, I can find the minimum perimeter.But in any case, let's proceed.So, for part 1, I think the correct expression is y = 2x + 8/x.Now, moving on to part 2(a): Prove that the function f(x) = x + 4/x is a decreasing function on (0, 2] and increasing on [2, ∞).Wait, but in part 1, I have y = 2x + 8/x, which is twice f(x). So, perhaps I need to analyze f(x) = x + 4/x, which is half the perimeter.So, perhaps the problem is considering f(x) = x + 4/x, and then in part 2(a), they want me to prove that f(x) is decreasing on (0, 2] and increasing on [2, ∞). Then, in part 2(b), find the x that minimizes the perimeter, which would correspond to the minimum of f(x) as well.So, perhaps I need to proceed with f(x) = x + 4/x.So, let's consider f(x) = x + 4/x.To analyze whether it's increasing or decreasing, I can take its derivative.So, f'(x) = derivative of x is 1, derivative of 4/x is -4/x².So, f'(x) = 1 - 4/x².Now, to find where the function is increasing or decreasing, we can look at the sign of f'(x).So, f'(x) = 1 - 4/x².Set f'(x) = 0 to find critical points.1 - 4/x² = 0So, 1 = 4/x²Multiply both sides by x²: x² = 4So, x = ±2. But since x is a length, it must be positive, so x = 2.So, the critical point is at x = 2.Now, to determine the intervals where f'(x) is positive or negative.Let's test intervals around x = 2.For x in (0, 2):Pick x = 1: f'(1) = 1 - 4/1 = 1 - 4 = -3 < 0. So, f(x) is decreasing on (0, 2).For x in (2, ∞):Pick x = 3: f'(3) = 1 - 4/9 ≈ 1 - 0.444 ≈ 0.556 > 0. So, f(x) is increasing on (2, ∞).Therefore, f(x) is decreasing on (0, 2] and increasing on [2, ∞). So, that proves part 2(a).Now, part 2(b): At what value of x is the length of the wire the shortest?Since f(x) is decreasing on (0, 2] and increasing on [2, ∞), the minimum occurs at x = 2.Therefore, the length of the wire is shortest when x = 2.But wait, let's think about this in terms of the perimeter.If f(x) = x + 4/x is half the perimeter, then the perimeter is 2*f(x). So, the perimeter is minimized when f(x) is minimized, which is at x = 2.So, the minimal perimeter occurs when x = 2, and then y = 4/x = 2. So, the rectangle is actually a square with sides 2 and 2.Therefore, the minimal perimeter is 2*(2 + 2) = 8.Wait, but in part 1, I had y = 2x + 8/x. So, if x = 2, then y = 2*2 + 8/2 = 4 + 4 = 8. So, that's correct.So, the minimal length of the wire is 8, occurring when x = 2.Therefore, the answer to part 2(b) is x = 2.But let me double-check everything.In part 1, I expressed the perimeter as y = 2x + 8/x, which is correct because the area is 4, so y = 4/x, and perimeter is 2*(x + y) = 2x + 8/x.In part 2(a), they refer to f(x) = x + 4/x, which is half the perimeter. So, if I analyze f(x), I find that it's decreasing on (0, 2] and increasing on [2, ∞), so the minimum occurs at x = 2.Therefore, the minimal perimeter is when x = 2, which makes the rectangle a square with sides 2 and 2, giving a perimeter of 8.So, everything checks out.Wait, but just to make sure, let me consider another value of x to see if the perimeter is indeed larger.For example, if x = 1, then y = 4, so perimeter is 2*(1 + 4) = 10, which is larger than 8.If x = 3, then y = 4/3 ≈ 1.333, so perimeter is 2*(3 + 1.333) ≈ 2*(4.333) ≈ 8.666, which is also larger than 8.So, yes, the minimal perimeter occurs at x = 2.Therefore, the answers are:1. y = 2x + 8/x2. (a) f(x) is decreasing on (0, 2] and increasing on [2, ∞) (b) The length of the wire is shortest when x = 2.But wait, in part 1, the problem says "express the total length of the wire, y, as a function of a side length x of the rectangle." So, I think my function y = 2x + 8/x is correct.But in part 2, they refer to f(x) = x + 4/x, which is half the perimeter. So, perhaps they are considering f(x) as a helper function to find the minimum.But in any case, the minimal perimeter occurs at x = 2, so that's the answer.I think I've covered all the steps and checked my work. I don't see any mistakes in my reasoning.

answer：Okay, so I have this problem about an arithmetic sequence. Let me try to understand it step by step. First, the problem says: Consider an arithmetic sequence {a_n} with a non-zero common difference. Given that a_3 = 7 and a_1 - 1, a_2 - 1, a_4 - 1 form a geometric sequence, find the value of a_10.Alright, let's break this down. An arithmetic sequence has a common difference, which means each term is the previous term plus some constant, right? So, if I denote the first term as a_1 and the common difference as d, then the nth term can be written as a_n = a_1 + (n - 1)d.Given that a_3 = 7, let's write that out. a_3 = a_1 + 2d = 7. So that's our first equation: a_1 + 2d = 7.Now, the other part says that a_1 - 1, a_2 - 1, a_4 - 1 form a geometric sequence. Hmm, okay. A geometric sequence has a common ratio, so each term is the previous term multiplied by some constant ratio, r.Let me denote the terms of the geometric sequence as follows:First term: a_1 - 1Second term: a_2 - 1Third term: a_4 - 1Since it's a geometric sequence, the ratio between the second and first term should be equal to the ratio between the third and second term. So, (a_2 - 1)/(a_1 - 1) = (a_4 - 1)/(a_2 - 1). Let me write that equation down:(a_2 - 1)/(a_1 - 1) = (a_4 - 1)/(a_2 - 1)Cross-multiplying, we get:(a_2 - 1)^2 = (a_1 - 1)(a_4 - 1)Okay, so now I need to express a_2 and a_4 in terms of a_1 and d.Since it's an arithmetic sequence:a_2 = a_1 + da_4 = a_1 + 3dSo, substituting these into the equation:(a_2 - 1)^2 = (a_1 - 1)(a_4 - 1)Becomes:(a_1 + d - 1)^2 = (a_1 - 1)(a_1 + 3d - 1)Let me expand both sides.First, the left side:(a_1 + d - 1)^2 = [ (a_1 - 1) + d ]^2 = (a_1 - 1)^2 + 2d(a_1 - 1) + d^2Right side:(a_1 - 1)(a_1 + 3d - 1) = (a_1 - 1)(a_1 - 1 + 3d) = (a_1 - 1)^2 + 3d(a_1 - 1)So, putting it all together:Left side: (a_1 - 1)^2 + 2d(a_1 - 1) + d^2Right side: (a_1 - 1)^2 + 3d(a_1 - 1)Subtracting right side from both sides:Left side - Right side = [ (a_1 - 1)^2 + 2d(a_1 - 1) + d^2 ] - [ (a_1 - 1)^2 + 3d(a_1 - 1) ] = 0Simplify:(a_1 - 1)^2 cancels out.2d(a_1 - 1) + d^2 - 3d(a_1 - 1) = 0Combine like terms:(2d - 3d)(a_1 - 1) + d^2 = 0Which is:(-d)(a_1 - 1) + d^2 = 0Factor out d:d[ - (a_1 - 1) + d ] = 0So, either d = 0 or - (a_1 - 1) + d = 0.But the problem says the common difference is non-zero, so d ≠ 0. Therefore:- (a_1 - 1) + d = 0Which simplifies to:- a_1 + 1 + d = 0So,- a_1 + d = -1Or,a_1 - d = 1So, that's our second equation: a_1 - d = 1.Earlier, we had the first equation: a_1 + 2d = 7.Now, we have a system of two equations:1) a_1 + 2d = 72) a_1 - d = 1Let me solve this system.From equation 2: a_1 = 1 + dSubstitute into equation 1:(1 + d) + 2d = 7Simplify:1 + 3d = 7Subtract 1:3d = 6Divide by 3:d = 2Now, substitute d = 2 into equation 2: a_1 - 2 = 1 => a_1 = 3So, the first term is 3, and the common difference is 2.Therefore, the arithmetic sequence is:a_1 = 3a_2 = 3 + 2 = 5a_3 = 5 + 2 = 7 (which matches the given)a_4 = 7 + 2 = 9And so on.Now, the problem asks for a_10.Since a_n = a_1 + (n - 1)d, so a_10 = 3 + (10 - 1)*2 = 3 + 9*2 = 3 + 18 = 21.Wait, let me double-check that.a_10 = a_1 + 9d = 3 + 9*2 = 3 + 18 = 21. Yes, that seems right.But just to make sure I didn't make any mistakes earlier, let me verify the geometric sequence part.Given a_1 -1 = 3 -1 = 2a_2 -1 = 5 -1 = 4a_4 -1 = 9 -1 = 8So, the sequence is 2, 4, 8.Is that a geometric sequence? Let's see.4 / 2 = 28 / 4 = 2Yes, the common ratio is 2, so it is a geometric sequence. Perfect.So, everything checks out.Therefore, the value of a_10 is 21.**Final Answer**The value of ( a_{10} ) is boxed{21}.