💡Okay, so I have this problem where I'm given that ( a + a^{-1} = 6 ), and I need to find the value of ( a - a^{-1} ). Hmm, let me think about how to approach this.First, I know that ( a^{-1} ) is the same as ( frac{1}{a} ), so the equation can be rewritten as ( a + frac{1}{a} = 6 ). That seems manageable. I need to find ( a - frac{1}{a} ). I remember that sometimes squaring both sides of an equation can help when dealing with expressions like this. Let me try that. If I square both sides of the equation ( a + frac{1}{a} = 6 ), I get:[(a + frac{1}{a})^2 = 6^2]Expanding the left side using the formula ( (x + y)^2 = x^2 + 2xy + y^2 ), I have:[a^2 + 2 cdot a cdot frac{1}{a} + left( frac{1}{a} right)^2 = 36]Simplifying each term, ( a cdot frac{1}{a} = 1 ), so the middle term becomes 2. Therefore, the equation becomes:[a^2 + 2 + frac{1}{a^2} = 36]Subtracting 2 from both sides gives:[a^2 + frac{1}{a^2} = 34]Okay, so now I have ( a^2 + frac{1}{a^2} = 34 ). But I need to find ( a - frac{1}{a} ). Maybe I can use a similar approach by squaring ( a - frac{1}{a} ).Let me denote ( x = a - frac{1}{a} ). Then, squaring both sides:[x^2 = left( a - frac{1}{a} right)^2]Expanding the right side using the formula ( (x - y)^2 = x^2 - 2xy + y^2 ), I get:[x^2 = a^2 - 2 cdot a cdot frac{1}{a} + frac{1}{a^2}]Simplifying each term, ( a cdot frac{1}{a} = 1 ), so the middle term becomes -2. Therefore:[x^2 = a^2 - 2 + frac{1}{a^2}]But from earlier, I know that ( a^2 + frac{1}{a^2} = 34 ). So substituting that in:[x^2 = 34 - 2 = 32]So, ( x^2 = 32 ). Taking the square root of both sides gives:[x = pm sqrt{32}]Simplifying ( sqrt{32} ), I know that ( 32 = 16 times 2 ), so:[sqrt{32} = sqrt{16 times 2} = sqrt{16} times sqrt{2} = 4sqrt{2}]Therefore, ( x = pm 4sqrt{2} ). Since ( x = a - frac{1}{a} ), this means:[a - frac{1}{a} = pm 4sqrt{2}]So, the value of ( a - a^{-1} ) is either ( 4sqrt{2} ) or ( -4sqrt{2} ). I think that's the answer. Let me just double-check my steps to make sure I didn't make any mistakes.Starting from ( a + frac{1}{a} = 6 ), I squared both sides correctly to get ( a^2 + 2 + frac{1}{a^2} = 36 ), then subtracted 2 to find ( a^2 + frac{1}{a^2} = 34 ). Then, I set ( x = a - frac{1}{a} ), squared it, and used the previously found value to get ( x^2 = 32 ), leading to ( x = pm 4sqrt{2} ). Everything seems to check out.I wonder if there's another way to approach this problem, maybe by solving for ( a ) first and then computing ( a - frac{1}{a} ). Let me try that.Starting with ( a + frac{1}{a} = 6 ), I can multiply both sides by ( a ) to eliminate the fraction:[a^2 + 1 = 6a]Rearranging terms gives a quadratic equation:[a^2 - 6a + 1 = 0]Using the quadratic formula ( a = frac{-b pm sqrt{b^2 - 4ac}}{2a} ) where ( a = 1 ), ( b = -6 ), and ( c = 1 ):[a = frac{6 pm sqrt{(-6)^2 - 4 cdot 1 cdot 1}}{2 cdot 1} = frac{6 pm sqrt{36 - 4}}{2} = frac{6 pm sqrt{32}}{2} = frac{6 pm 4sqrt{2}}{2} = 3 pm 2sqrt{2}]So, ( a = 3 + 2sqrt{2} ) or ( a = 3 - 2sqrt{2} ). Now, let's compute ( a - frac{1}{a} ) for each case.First, take ( a = 3 + 2sqrt{2} ). Then, ( frac{1}{a} ) is:[frac{1}{3 + 2sqrt{2}} = frac{3 - 2sqrt{2}}{(3 + 2sqrt{2})(3 - 2sqrt{2})} = frac{3 - 2sqrt{2}}{9 - (2sqrt{2})^2} = frac{3 - 2sqrt{2}}{9 - 8} = 3 - 2sqrt{2}]So, ( a - frac{1}{a} = (3 + 2sqrt{2}) - (3 - 2sqrt{2}) = 4sqrt{2} ).Next, take ( a = 3 - 2sqrt{2} ). Then, ( frac{1}{a} ) is:[frac{1}{3 - 2sqrt{2}} = frac{3 + 2sqrt{2}}{(3 - 2sqrt{2})(3 + 2sqrt{2})} = frac{3 + 2sqrt{2}}{9 - (2sqrt{2})^2} = frac{3 + 2sqrt{2}}{9 - 8} = 3 + 2sqrt{2}]So, ( a - frac{1}{a} = (3 - 2sqrt{2}) - (3 + 2sqrt{2}) = -4sqrt{2} ).This confirms the earlier result that ( a - a^{-1} = pm 4sqrt{2} ). Therefore, both methods lead to the same conclusion, which makes me more confident that the answer is correct.I think I've covered all the necessary steps and checked my work by solving for ( a ) explicitly. It seems solid. I don't see any mistakes in my reasoning, so I'm ready to present the final answer.

💡Okay, so I have this sequence problem here. The first term is 3243, and each next term is the sum of the squares of the digits of the previous term plus 2. I need to find the 1001st term. Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, let me write down the first term: 3243. Now, to find the second term, I need to take each digit of 3243, square it, add those squares together, and then add 2. Let me do that.The digits of 3243 are 3, 2, 4, and 3. Squaring each of these: 3 squared is 9, 2 squared is 4, 4 squared is 16, and 3 squared is 9 again. So adding those up: 9 + 4 + 16 + 9. Let me calculate that: 9 + 4 is 13, 13 + 16 is 29, and 29 + 9 is 38. Then, I need to add 2, so 38 + 2 is 40. So the second term is 40.Wait, hold on, the assistant's initial response had a different calculation. They got 52 as the second term. Did I do something wrong? Let me check again. The digits are 3, 2, 4, 3. Squaring each: 3²=9, 2²=4, 4²=16, 3²=9. Adding them: 9 + 4 + 16 + 9. That's 9+4=13, 13+16=29, 29+9=38. Then 38 + 2=40. Hmm, so I get 40, but the assistant got 52. Maybe I misread the problem? Let me check.Wait, the problem says each succeeding term is the sum of the squares of the digits of the previous term plus 2. So, for 3243, the digits are 3, 2, 4, 3. Squared: 9, 4, 16, 9. Sum is 38, plus 2 is 40. So I think I'm correct. Maybe the assistant made a mistake there. Let me proceed with my calculation.So, term 1: 3243, term 2: 40. Now, term 3: take 40, digits are 4 and 0. Squaring each: 4²=16, 0²=0. Sum is 16 + 0 = 16, plus 2 is 18. So term 3 is 18.Term 4: 18. Digits are 1 and 8. Squaring: 1²=1, 8²=64. Sum is 1 + 64 = 65, plus 2 is 67. Term 4 is 67.Term 5: 67. Digits are 6 and 7. Squaring: 6²=36, 7²=49. Sum is 36 + 49 = 85, plus 2 is 87. Term 5 is 87.Term 6: 87. Digits are 8 and 7. Squaring: 8²=64, 7²=49. Sum is 64 + 49 = 113, plus 2 is 115. Term 6 is 115.Term 7: 115. Digits are 1, 1, and 5. Squaring: 1²=1, 1²=1, 5²=25. Sum is 1 + 1 + 25 = 27, plus 2 is 29. Term 7 is 29.Term 8: 29. Digits are 2 and 9. Squaring: 2²=4, 9²=81. Sum is 4 + 81 = 85, plus 2 is 87. Term 8 is 87.Wait a minute, term 8 is 87, which was term 5. So now I see a repetition: 87, 115, 29, 87, 115, 29, and so on. So the sequence is entering a cycle here. Let me list the terms again to see the cycle clearly.Term 1: 3243Term 2: 40Term 3: 18Term 4: 67Term 5: 87Term 6: 115Term 7: 29Term 8: 87Term 9: 115Term 10: 29Term 11: 87... and so on.So, starting from term 5, which is 87, the sequence cycles through 87, 115, 29, and then back to 87. So the cycle length is 3: 87, 115, 29, 87, 115, 29, etc.Wait, so the cycle is 87, 115, 29, and then repeats. So the period is 3 terms. That means every 3 terms after term 5, it repeats.Now, I need to find the 1001st term. Let me see how many terms are in the cycle and where 1001 falls.First, let's note that the cycle starts at term 5. So terms 5, 6, 7 are the first cycle: 87, 115, 29.Then terms 8, 9, 10: 87, 115, 29.And so on.So, starting from term 5, each cycle is 3 terms. So, to find the 1001st term, I need to see how many terms are after term 5, which is 1001 - 4 = 997 terms.Wait, because term 1 is the first term, so term 5 is the fifth term. So from term 5 to term 1001, that's 1001 - 4 = 997 terms.Now, since the cycle is 3 terms, I can divide 997 by 3 to see how many complete cycles there are and what the remainder is.Let me calculate 997 divided by 3.3 goes into 997 how many times? 3*332=996, so 332 cycles with a remainder of 1.So, 997 = 3*332 + 1.That means after 332 complete cycles (each of 3 terms), there is 1 extra term.So, starting from term 5, which is 87, the first term of the cycle, the 1st term in the cycle is 87, the 2nd is 115, the 3rd is 29, then back to 87, and so on.Since the remainder is 1, the 997th term after term 4 (which is term 1001) will be the first term in the cycle, which is 87.Wait, but hold on, let me double-check that.Wait, term 5 is the first term of the cycle, which is 87.So, term 5: 87 (cycle position 1)term 6: 115 (cycle position 2)term 7: 29 (cycle position 3)term 8: 87 (cycle position 1)term 9: 115 (cycle position 2)term 10: 29 (cycle position 3)So, the cycle positions are 1,2,3,1,2,3,...So, if I have 997 terms after term 4, which is term 5 to term 1001, and 997 divided by 3 is 332 with a remainder of 1, that means 332 full cycles (each of 3 terms) account for 996 terms, and then 1 more term.So, starting from term 5 (cycle position 1), adding 996 terms brings us to term 5 + 996 = term 1001. Wait, no, term 5 is the first term of the cycle, so term 5 is position 1, term 6 is position 2, term 7 is position 3, term 8 is position 1, etc.So, if I have 997 terms starting from term 5, the 997th term would be term 5 + 997 -1 = term 1001.Wait, maybe I'm overcomplicating. Let me think differently.The cycle starts at term 5. So term 5 is the first term of the cycle, term 6 is the second, term 7 is the third, term 8 is the first again, etc.So, to find term 1001, I need to find how many terms after term 4, which is 1001 - 4 = 997 terms.Now, 997 divided by 3 is 332 with a remainder of 1. So, 332 full cycles (each of 3 terms) account for 996 terms, and then 1 more term.So, starting from term 5, which is cycle position 1, adding 996 terms brings us to term 5 + 996 = term 1001. But wait, term 5 is the first term of the cycle, so term 5 is position 1, term 6 is position 2, term 7 is position 3, term 8 is position 1, etc.So, if I have 997 terms starting from term 5, the 997th term would be term 5 + 997 -1 = term 1001.Wait, maybe I'm overcomplicating. Let me think of it as term 5 is the first term of the cycle, so term 5 corresponds to n=1 in the cycle, term 6 is n=2, term 7 is n=3, term 8 is n=1, etc.So, term number = 5 + (n-1), where n is the position in the cycle.So, to find term 1001, we can express it as 5 + (k-1), where k is the position in the cycle.So, 1001 = 5 + (k-1) => k = 1001 - 4 = 997.So, k=997. Now, since the cycle length is 3, we can find the position in the cycle by 997 mod 3.997 divided by 3 is 332 with a remainder of 1, because 3*332=996, and 997-996=1.So, the remainder is 1, which means term 1001 corresponds to the first term in the cycle, which is 87.Wait, but earlier I thought the cycle was 87, 115, 29, so the first term is 87, second is 115, third is 29.So, if the remainder is 1, it's 87, remainder 2 is 115, remainder 0 (if divisible) would be 29.So, term 1001 is 87.But wait, in the initial calculation, the assistant got 51 as the answer, but I'm getting 87. Did I make a mistake somewhere?Wait, let me go back to the beginning. Maybe I messed up the initial terms.Wait, the first term is 3243.Term 1: 3243Term 2: sum of squares of digits of 3243 plus 2.Digits: 3,2,4,3.Squares: 9,4,16,9.Sum: 9+4=13, 13+16=29, 29+9=38. 38+2=40. So term 2 is 40.Term 3: sum of squares of digits of 40 plus 2.Digits:4,0.Squares:16,0.Sum:16+0=16. 16+2=18. Term 3:18.Term 4: sum of squares of digits of 18 plus 2.Digits:1,8.Squares:1,64.Sum:1+64=65. 65+2=67. Term 4:67.Term 5: sum of squares of digits of 67 plus 2.Digits:6,7.Squares:36,49.Sum:36+49=85. 85+2=87. Term 5:87.Term 6: sum of squares of digits of 87 plus 2.Digits:8,7.Squares:64,49.Sum:64+49=113. 113+2=115. Term 6:115.Term 7: sum of squares of digits of 115 plus 2.Digits:1,1,5.Squares:1,1,25.Sum:1+1+25=27. 27+2=29. Term 7:29.Term 8: sum of squares of digits of 29 plus 2.Digits:2,9.Squares:4,81.Sum:4+81=85. 85+2=87. Term 8:87.So, term 8 is 87, which was term 5. So the cycle is 87,115,29,87,115,29,...So, starting from term 5, the cycle is 3 terms:87,115,29.So, term 5:87 (cycle position 1)term 6:115 (cycle position 2)term 7:29 (cycle position 3)term 8:87 (cycle position 1)term 9:115 (cycle position 2)term 10:29 (cycle position 3)and so on.So, to find term 1001, we need to see how many terms after term 4, which is 1001-4=997 terms.Now, 997 divided by 3 is 332 with a remainder of 1, because 3*332=996, and 997-996=1.So, the remainder is 1, which means term 1001 is the first term in the cycle, which is 87.Wait, but the assistant's initial response said the answer was 51, but I'm getting 87. Did I make a mistake?Wait, let me check the assistant's initial response again.They said:- First term (a0) is 3243.- Next term (a1) is 3² + 2⁴ + 4² + 3² + 2 = 9 + 16 + 16 + 9 + 2 = 52.Wait, hold on, that's different from my calculation. They added 2⁴ instead of 2². That's a mistake.Because the rule is sum of the squares of the digits plus 2. So for 3243, digits are 3,2,4,3. So squares are 9,4,16,9. Sum is 38, plus 2 is 40, not 52.So, the assistant made a mistake in their initial calculation. They incorrectly calculated 2⁴ instead of 2². So their entire sequence was wrong from the start.So, in reality, the correct sequence is:Term 1:3243Term 2:40Term 3:18Term 4:67Term 5:87Term 6:115Term 7:29Term 8:87And so on, with the cycle starting at term 5:87,115,29.Therefore, the correct answer should be 87, not 51.Wait, but let me make sure I didn't make any other mistakes.Let me recalculate the terms step by step.Term 1:3243Term 2: sum of squares of digits of 3243 plus 2.Digits:3,2,4,3.Squares:9,4,16,9.Sum:9+4=13, 13+16=29, 29+9=38. 38+2=40. So term 2:40.Term 3: sum of squares of digits of 40 plus 2.Digits:4,0.Squares:16,0.Sum:16+0=16. 16+2=18. Term 3:18.Term 4: sum of squares of digits of 18 plus 2.Digits:1,8.Squares:1,64.Sum:1+64=65. 65+2=67. Term 4:67.Term 5: sum of squares of digits of 67 plus 2.Digits:6,7.Squares:36,49.Sum:36+49=85. 85+2=87. Term 5:87.Term 6: sum of squares of digits of 87 plus 2.Digits:8,7.Squares:64,49.Sum:64+49=113. 113+2=115. Term 6:115.Term 7: sum of squares of digits of 115 plus 2.Digits:1,1,5.Squares:1,1,25.Sum:1+1+25=27. 27+2=29. Term 7:29.Term 8: sum of squares of digits of 29 plus 2.Digits:2,9.Squares:4,81.Sum:4+81=85. 85+2=87. Term 8:87.So, term 8 is 87, which was term 5. So the cycle is 87,115,29,87,115,29,...So, starting from term 5, the cycle is 3 terms:87,115,29.Therefore, term 5:87 (cycle position 1)term 6:115 (cycle position 2)term 7:29 (cycle position 3)term 8:87 (cycle position 1)term 9:115 (cycle position 2)term 10:29 (cycle position 3)and so on.So, to find term 1001, we need to see how many terms after term 4, which is 1001-4=997 terms.Now, 997 divided by 3 is 332 with a remainder of 1, because 3*332=996, and 997-996=1.So, the remainder is 1, which means term 1001 is the first term in the cycle, which is 87.Therefore, the 1001st term is 87.Wait, but the assistant's initial response said 51. So, I think the assistant made a mistake in their calculation, probably because they incorrectly calculated the squares of the digits in the first step.So, in conclusion, the correct answer should be 87, not 51.

💡Okay, so I have this problem here about using mathematical induction to prove that the sum of the first n odd numbers is equal to n squared. The equation given is 1 + 3 + 5 + ... + (2n - 1) = n², where n is a positive integer. The question is asking about the step in the induction process where we assume the equation holds for n = k and then what we should get for n = k + 1. The options are A, B, C, D, each giving a different expression for the sum up to (2k + 1).Alright, let me recall how mathematical induction works. It has two main steps: the base case and the inductive step. The base case is usually proving the statement holds for n = 1. Then, assuming it holds for some arbitrary positive integer k, we prove it holds for k + 1.So, for the base case, when n = 1, the left side is just 1, and the right side is 1², which is also 1. So, the base case checks out.Now, moving on to the inductive step. We assume that the equation holds for n = k. That means we assume that 1 + 3 + 5 + ... + (2k - 1) = k². This is our induction hypothesis.Next, we need to show that if the equation holds for n = k, then it also holds for n = k + 1. That is, we need to show that 1 + 3 + 5 + ... + (2k - 1) + (2(k + 1) - 1) = (k + 1)².Wait, let me make sure I get the term right. For n = k + 1, the last term in the sum should be 2(k + 1) - 1, which simplifies to 2k + 2 - 1, so 2k + 1. So, the sum becomes 1 + 3 + 5 + ... + (2k - 1) + (2k + 1).Given that we assumed the sum up to (2k - 1) is k², we can substitute that in. So, the sum up to (2k + 1) would be k² + (2k + 1). Let me compute that: k² + 2k + 1. Hmm, that looks familiar. That factors into (k + 1)². So, that shows that if the equation holds for n = k, it also holds for n = k + 1.Looking back at the options provided, option B says that 1 + 3 + 5 + ... + (2k + 1) = (k + 1)², which is exactly what we derived. So, option B is the correct one.Wait, let me double-check. If I choose option A, it says the sum equals k², but we know that when we add (2k + 1) to k², we get (k + 1)², not k². So, A is incorrect. Option C says it equals (k + 2)², which is too big because we only added one term. Similarly, option D is even larger, so that's not right either. So, yeah, B is definitely the correct answer.I think I got it. The key was understanding how the induction step works: assuming it's true for k, then showing it's true for k + 1 by adding the next term and simplifying. It's a common technique in proofs, so it's good to get comfortable with it.Just to recap, the induction hypothesis is 1 + 3 + 5 + ... + (2k - 1) = k². Then, adding the next term, which is (2k + 1), gives us k² + (2k + 1) = (k + 1)². Therefore, the statement holds for k + 1, completing the inductive step.I don't think I made any mistakes here. It all follows logically. The only thing that could trip someone up is miscounting the terms or miscalculating the algebra, but as long as you're careful, it's straightforward.So, yeah, I feel confident that option B is the right choice here. It correctly represents the sum for n = k + 1 based on the induction hypothesis.**Final Answer**boxed{B}

💡First, calculate the total number of eggs distributed to the adults. There are three adults, and each received 3 eggs:3 adults * 3 eggs/adult = 9 eggsNext, determine the total number of eggs prepared for the trip:3 dozen eggs = 3 * 12 eggs = 36 eggsSubtract the eggs given to the adults to find out how many eggs are left for the children:36 eggs - 9 eggs = 27 eggsDistribute the eggs to the girls. There are 7 girls, and each received 1 egg:7 girls * 1 egg/girl = 7 eggsSubtract the eggs given to the girls from the remaining eggs:27 eggs - 7 eggs = 20 eggsThese 20 eggs are to be distributed among the 10 boys:20 eggs / 10 boys = 2 eggs/boyFinally, find out how many more eggs each boy received compared to each girl:2 eggs/boy - 1 egg/girl = 1 eggSo, each boy received 1 egg more than each girl.

💡First, identify the total number of students and the number of sophomores.Next, determine the proportion of sophomores in the entire student population.Then, apply this proportion to the total number of students selected for the physical fitness test.Finally, calculate the number of sophomores selected by multiplying the proportion by the total number of students selected.

💡First, I need to determine the capacity of each type of cupboard. Let's denote the number of glasses the tall cupboard can hold as ( T ).The wide cupboard can hold twice as many glasses as the tall cupboard, so it holds ( 2T ) glasses.The narrow cupboard can hold 15 glasses, with the glasses divided equally among three shelves. This means each shelf holds ( frac{15}{3} = 5 ) glasses.However, one shelf in the narrow cupboard broke, leaving only two functional shelves. Therefore, the narrow cupboard now holds ( 2 times 5 = 10 ) glasses.Since all cupboards are full, the total number of glasses is the sum of the glasses in the tall, wide, and narrow cupboards:[T + 2T + 10 = 3T + 10]Given that the narrow cupboard originally held 15 glasses divided equally among three shelves, each shelf holds 5 glasses. This implies that the tall cupboard also holds 5 glasses.Therefore, the tall cupboard currently holds (boxed{5}) glasses.