answer：Okay, so I have this problem here where there's an equilateral triangle ABC with side length (2sqrt{2}), and all its vertices lie on the surface of a sphere. The volume of the sphere is given as (4sqrt{3}pi). I need to find the maximum distance from a point on the sphere to the plane ABC.Hmm, let me start by recalling some formulas. First, the volume of a sphere is given by (frac{4}{3}pi R^3), where (R) is the radius. Since the volume is (4sqrt{3}pi), I can set up the equation:[frac{4}{3}pi R^3 = 4sqrt{3}pi]I can cancel out the (pi) from both sides, so that leaves me with:[frac{4}{3} R^3 = 4sqrt{3}]Now, I can multiply both sides by (frac{3}{4}) to solve for (R^3):[R^3 = 4sqrt{3} times frac{3}{4} = 3sqrt{3}]Wait, that doesn't seem right. Let me check my steps again. Oh, no, actually, if I have (frac{4}{3} R^3 = 4sqrt{3}), then multiplying both sides by (frac{3}{4}) gives:[R^3 = 4sqrt{3} times frac{3}{4} = 3sqrt{3}]Hmm, so (R^3 = 3sqrt{3}). To find (R), I need to take the cube root of (3sqrt{3}). Let me express (3sqrt{3}) as (3^{3/2}), so:[R = (3^{3/2})^{1/3} = 3^{(3/2 times 1/3)} = 3^{1/2} = sqrt{3}]Okay, so the radius (R) of the sphere is (sqrt{3}). That seems correct.Now, moving on to the equilateral triangle ABC with side length (2sqrt{2}). Since it's an equilateral triangle, all sides are equal, and all angles are 60 degrees. The triangle lies on the surface of the sphere, so it must be a spherical triangle, but since all its vertices are on the sphere, the plane of the triangle is at some distance from the center of the sphere.I need to find the maximum distance from a point on the sphere to the plane ABC. To do this, I think I need to find the distance from the center of the sphere to the plane ABC, and then the maximum distance from a point on the sphere to the plane would be the sum of this distance and the radius of the sphere. Alternatively, if the distance from the center to the plane is (d), then the maximum distance from a point on the sphere to the plane would be (R + d) if the point is on the opposite side of the sphere from the plane, or (R - d) if it's on the same side. Since we're looking for the maximum, it should be (R + d).So, first, I need to find the distance (d) from the center of the sphere to the plane ABC. To find this, I can use the formula relating the radius of the sphere, the radius of the circumscribed circle of the triangle, and the distance (d). The formula is:[R^2 = r^2 + d^2]Where (r) is the radius of the circumscribed circle of triangle ABC. So, if I can find (r), I can solve for (d).Since ABC is an equilateral triangle, the radius (r) of its circumscribed circle can be found using the formula:[r = frac{a}{sqrt{3}}]Where (a) is the side length of the triangle. Plugging in (a = 2sqrt{2}):[r = frac{2sqrt{2}}{sqrt{3}} = frac{2sqrt{6}}{3}]Okay, so (r = frac{2sqrt{6}}{3}). Now, plugging this back into the equation (R^2 = r^2 + d^2):[(sqrt{3})^2 = left(frac{2sqrt{6}}{3}right)^2 + d^2]Calculating each term:[3 = left(frac{4 times 6}{9}right) + d^2][3 = frac{24}{9} + d^2][3 = frac{8}{3} + d^2]Subtracting (frac{8}{3}) from both sides:[d^2 = 3 - frac{8}{3} = frac{9}{3} - frac{8}{3} = frac{1}{3}]So, (d = sqrt{frac{1}{3}} = frac{sqrt{3}}{3}).Therefore, the distance from the center of the sphere to the plane ABC is (frac{sqrt{3}}{3}). Now, to find the maximum distance from a point on the sphere to the plane ABC, I need to consider the point on the sphere that is farthest from the plane. This point would be in the direction away from the plane, so the distance would be the sum of the distance from the center to the plane and the radius of the sphere.So, the maximum distance is:[R + d = sqrt{3} + frac{sqrt{3}}{3} = frac{3sqrt{3}}{3} + frac{sqrt{3}}{3} = frac{4sqrt{3}}{3}]Let me just double-check my steps to make sure I didn't make any mistakes.1. Calculated the radius (R) of the sphere correctly: (R = sqrt{3}).2. Found the radius (r) of the circumscribed circle of the equilateral triangle: (r = frac{2sqrt{6}}{3}).3. Applied the formula (R^2 = r^2 + d^2) to find (d = frac{sqrt{3}}{3}).4. Added (R) and (d) to get the maximum distance: (frac{4sqrt{3}}{3}).Everything seems to check out. So, the maximum distance from a point on the sphere to the plane ABC is (frac{4sqrt{3}}{3}).**Final Answer**The maximum distance from a point on the sphere to plane ABC is boxed{dfrac{4sqrt{3}}{3}}.

answer：Alright, so I have this problem here about a stick that's being cut in half multiple times, and I need to figure out its length after the fifth cut. Let me try to understand what's going on here.First off, the stick starts with a length of 1 unit. Every time it's cut in half, its length becomes half of what it was before. So, after the first cut, it should be 1/2. After the second cut, it should be 1/4, right? Because half of 1/2 is 1/4. Okay, that makes sense so far.Let me write this down to keep track:- After 0 cuts: 1- After 1 cut: 1/2- After 2 cuts: 1/4- After 3 cuts: 1/8- After 4 cuts: 1/16- After 5 cuts: 1/32Wait a minute, so each time we're multiplying by 1/2. So, after n cuts, the length should be 1/(2^n). That seems straightforward. So, after 5 cuts, it should be 1/(2^5), which is 1/32. Hmm, that seems pretty clear.But let me double-check to make sure I'm not missing anything. The problem says the stick is cut in half each time. Does that mean we're always cutting the entire stick, or are we cutting one of the halves each time? I think it means cutting the entire stick each time, so each cut halves the current length.So, starting with 1:1st cut: 1 * 1/2 = 1/22nd cut: 1/2 * 1/2 = 1/43rd cut: 1/4 * 1/2 = 1/84th cut: 1/8 * 1/2 = 1/165th cut: 1/16 * 1/2 = 1/32Yep, that's consistent. So, after five cuts, the length is 1/32, which is 1/(2^5). Looking at the options:A: 1 - 1/2^5B: 1 - 1/2^6C: 1/2^6D: 1/2^5So, D is 1/2^5, which is exactly what I got. So, D should be the correct answer.But wait, let me think again. Sometimes, these problems can be tricky. Maybe they're asking about something else, like the total length of all the pieces after five cuts? But no, the question specifically says "the length of the stick after the 5th cut." So, it's referring to the remaining piece after each cut, not the total length of all pieces.If it were about the total length, it would be a different story. For example, after each cut, you have two pieces, each half the length of the original. So, the total length remains 1, but the number of pieces increases. But that's not what the question is asking.So, to clarify:- After each cut, the stick is cut into two equal parts, each half the length of the previous stick.- The question is asking for the length of the stick after the fifth cut, which I interpret as the length of one of the resulting pieces after the fifth cut.Therefore, after five cuts, each piece would be 1/(2^5) = 1/32 in length.Alternatively, if we were to consider the total length of all pieces after five cuts, it would still be 1, since we're just breaking the stick into smaller pieces without changing the total length. But again, the question is specifically about the length after the fifth cut, which I think refers to the length of the resulting piece.Just to make sure, let's consider the process step by step:1. Start with a stick of length 1.2. First cut: divides the stick into two pieces, each of length 1/2.3. Second cut: takes one of those halves and cuts it again into two pieces, each of length 1/4.4. Third cut: takes one of those quarters and cuts it into two pieces, each of length 1/8.5. Fourth cut: takes one of those eighths and cuts it into two pieces, each of length 1/16.6. Fifth cut: takes one of those sixteenths and cuts it into two pieces, each of length 1/32.So, after five cuts, the smallest piece is 1/32. Therefore, the length of the stick after the fifth cut is 1/32, which is 1/(2^5).Therefore, the correct answer is D: 1/2^5.I think that's solid. I don't see any mistakes in my reasoning. Each cut halves the length of the piece we're focusing on, and after five such cuts, it's 1/32. So, D is the right choice.

answer：First, I need to add the whole number 217 to the decimal numbers 2.017, 0.217, and 2.0017.I'll start by adding 217 and 2.017. This gives me 219.017.Next, I'll add 0.217 to 219.017, resulting in 219.234.Finally, I'll add 2.0017 to 219.234, which totals 221.2357.Therefore, the sum of 217, 2.017, 0.217, and 2.0017 is 221.2357.

answer：Okay, so I have this problem about a sequence defined by a recurrence relation, and I need to find a constant c and then figure out the general term of the sequence. Let me try to break this down step by step.First, the sequence is given as {a_n}, with a_1 = 2. Then, each subsequent term is defined by a_{n+1} = a_n + c*n, where c is a constant. That means each term is the previous term plus c times n. So, for example, a_2 would be a_1 + c*1, which is 2 + c. Similarly, a_3 would be a_2 + c*2, which is (2 + c) + 2c = 2 + 3c. Got that.Now, the problem also says that a_1, a_2, a_3 form a geometric sequence with a common ratio not equal to 1. A geometric sequence means that each term is a multiple of the previous term by a common ratio, say r. So, if a_1, a_2, a_3 are in geometric progression, then a_2 / a_1 = a_3 / a_2 = r, and r ≠ 1.So, let's write that out. From the geometric sequence condition, we have:a_2 / a_1 = a_3 / a_2Substituting the expressions we have for a_2 and a_3:(2 + c) / 2 = (2 + 3c) / (2 + c)Now, I can cross-multiply to solve for c. Let's do that:(2 + c)^2 = 2*(2 + 3c)Expanding the left side:(2 + c)^2 = 4 + 4c + c^2And the right side:2*(2 + 3c) = 4 + 6cSo, setting them equal:4 + 4c + c^2 = 4 + 6cSubtracting 4 from both sides:4c + c^2 = 6cSubtracting 6c from both sides:c^2 - 2c = 0Factor out c:c(c - 2) = 0So, c = 0 or c = 2.But wait, the problem says the common ratio is not equal to 1. If c = 0, then a_2 = 2 + 0 = 2, and a_3 = 2 + 0 = 2. So, all terms would be 2, which is a geometric sequence with ratio 1. But that's not allowed because the common ratio must not be 1. Therefore, c = 0 is invalid.So, the only valid solution is c = 2.Alright, so part I is solved: c = 2.Now, moving on to part II: finding the general term formula for {a_n}.We know that a_{n+1} = a_n + c*n, and c = 2. So, the recurrence relation becomes:a_{n+1} = a_n + 2nWe can write this as:a_{n+1} - a_n = 2nThis is a linear recurrence relation. To find the general term, we can solve this recurrence.First, let's write out the first few terms to see the pattern:a_1 = 2a_2 = a_1 + 2*1 = 2 + 2 = 4a_3 = a_2 + 2*2 = 4 + 4 = 8a_4 = a_3 + 2*3 = 8 + 6 = 14a_5 = a_4 + 2*4 = 14 + 8 = 22Hmm, so the terms are 2, 4, 8, 14, 22,...Looking at the differences between terms:a_2 - a_1 = 2a_3 - a_2 = 4a_4 - a_3 = 6a_5 - a_4 = 8So, the differences themselves form an arithmetic sequence with a common difference of 2. That makes sense because the recurrence is a_{n+1} - a_n = 2n, which is linear in n.To find the general term, we can use the method of summing the differences.We have:a_n = a_1 + sum_{k=1}^{n-1} (a_{k+1} - a_k)Which is:a_n = 2 + sum_{k=1}^{n-1} 2kSimplify the sum:sum_{k=1}^{n-1} 2k = 2 * sum_{k=1}^{n-1} k = 2 * [ (n-1)n / 2 ] = (n-1)nSo, putting it all together:a_n = 2 + n(n - 1)Simplify:a_n = n^2 - n + 2Let me check this formula with the terms I calculated earlier.For n=1: 1^2 -1 +2 = 1 -1 +2 = 2 ✔️For n=2: 4 -2 +2 = 4 ✔️For n=3: 9 -3 +2 = 8 ✔️For n=4: 16 -4 +2 = 14 ✔️For n=5: 25 -5 +2 = 22 ✔️Looks good.So, the general term is a_n = n^2 - n + 2.**Final Answer**(I) The value of ( c ) is (boxed{2}).(II) The general term formula for the sequence is (boxed{a_n = n^2 - n + 2}).

answer：Alright, so I've got this triangle problem here, and I need to solve two parts. Let me take it step by step.First, the problem says in triangle ABC, the sides opposite angles A, B, and C are a, b, and c respectively. It also gives an equation: (b)/(cos B) = (3c - a)/(cos A). Part (I) asks: If a = sqrt(2) sin A, find b.Okay, so I need to find the length of side b given that a is sqrt(2) times sin A. Hmm, I remember the Law of Sines relates sides and angles in a triangle. It states that a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle. Maybe that can help here.Given the equation (b)/(cos B) = (3c - a)/(cos A), I can try to express everything in terms of sines and cosines using the Law of Sines. Let me write down the Law of Sines for sides a, b, and c:a = 2R sin A b = 2R sin B c = 2R sin CSo, substituting these into the given equation:(2R sin B)/(cos B) = (3*(2R sin C) - 2R sin A)/(cos A)Simplify both sides by dividing numerator and denominator by 2R:(sin B)/(cos B) = (3 sin C - sin A)/(cos A)So, tan B = (3 sin C - sin A)/cos AHmm, that's interesting. Maybe I can manipulate this equation further. Let me write it as:sin B / cos B = (3 sin C - sin A)/cos ACross-multiplying:sin B cos A = (3 sin C - sin A) cos BExpanding the right side:sin B cos A = 3 sin C cos B - sin A cos BLet me bring all terms to one side:sin B cos A + sin A cos B = 3 sin C cos BWait, the left side looks familiar. Isn't that the sine of (A + B)? Because sin(A + B) = sin A cos B + cos A sin B. So,sin(A + B) = 3 sin C cos BBut in a triangle, A + B + C = π, so A + B = π - C. Therefore, sin(A + B) = sin(π - C) = sin C.So, substituting back:sin C = 3 sin C cos BAssuming sin C ≠ 0 (which it can't be in a triangle because angles are between 0 and π), we can divide both sides by sin C:1 = 3 cos BSo, cos B = 1/3. Therefore, sin B = sqrt(1 - (1/3)^2) = sqrt(8/9) = 2√2/3.Alright, so now we have sin B. Let's get back to the given information: a = sqrt(2) sin A.From the Law of Sines, a = 2R sin A, so 2R sin A = sqrt(2) sin A. If sin A ≠ 0, which it isn't in a triangle, we can divide both sides by sin A:2R = sqrt(2) R = sqrt(2)/2So, the radius R is sqrt(2)/2. Now, since b = 2R sin B, let's compute b:b = 2*(sqrt(2)/2)*sin B = sqrt(2)*(2√2/3) = sqrt(2)*(2√2)/3Multiplying sqrt(2) and 2√2: sqrt(2)*2√2 = 2*(sqrt(2)*sqrt(2)) = 2*2 = 4So, b = 4/3.Okay, that seems solid. Let me just recap:1. Started with the given equation and used Law of Sines to express sides in terms of sines.2. Simplified the equation to find that cos B = 1/3.3. Found sin B using Pythagorean identity.4. Used the given a = sqrt(2) sin A to find R.5. Finally, computed b using R and sin B.Part (I) done. Now, moving on to Part (II):Given b = 3 and the area of triangle ABC is 2√2, find a + c.Alright, so area is given by (1/2)*a*c*sin B. We know sin B from earlier, which is 2√2/3. So:Area = (1/2)*a*c*(2√2/3) = (a*c*√2)/3 = 2√2So, (a*c*√2)/3 = 2√2Multiply both sides by 3:a*c*√2 = 6√2Divide both sides by √2:a*c = 6So, the product of a and c is 6.We also know from the Law of Cosines that:b² = a² + c² - 2ac cos BWe know b = 3, cos B = 1/3, and ac = 6.So, plug in the values:3² = a² + c² - 2*6*(1/3)Simplify:9 = a² + c² - 4So, a² + c² = 13But we need to find a + c. Hmm, I remember that (a + c)² = a² + 2ac + c². We know a² + c² = 13 and ac = 6, so:(a + c)² = 13 + 2*6 = 13 + 12 = 25Therefore, a + c = sqrt(25) = 5Since sides are positive, a + c = 5.Let me verify that again:1. Area formula gave us a*c = 6.2. Law of Cosines gave us a² + c² = 13.3. Then, (a + c)² = a² + 2ac + c² = 13 + 12 = 25.4. So, a + c = 5.That seems correct.**Final Answer**(I) boxed{dfrac{4}{3}} (II) boxed{5}

answer：First, I recognize that the given ratio (frac{x}{y} = frac{7}{2}) implies that (x = 7a) and (y = 2a) for some common variable (a).Next, I substitute these expressions into the fraction (frac{x - 2y}{y}):[frac{x - 2y}{y} = frac{7a - 2(2a)}{2a} = frac{7a - 4a}{2a} = frac{3a}{2a}]Finally, I simplify the fraction by canceling out the common factor (a), resulting in:[frac{3}{2}]Therefore, the value of (frac{x - 2y}{y}) is (boxed{frac{3}{2}}).