answer:To find out how many placards each person collects, we need to divide the total number of placards by the number of people who have entered the stadium. The total number of placards is 4634. The number of people who have entered the stadium is 2317. So, each person collects: 4634 placards ÷ 2317 people = 2 placards per person (rounded down to the nearest whole number, as you can't collect a fraction of a placard). Therefore, each person collects boxed{2} placards.

answer:# Solution: Part 1: Finding the General Formula for {a_{n}} Given that S_{3}=7 and a_{3}, 3a_{2}, a_{4} form an arithmetic sequence, we can set up the following equations based on the properties of geometric and arithmetic sequences: 1. For the sum of the first three terms of a geometric sequence, we have: [S_{3} = a_{1}(1 + q + q^{2}) = 7] 2. Since a_{3}, 3a_{2}, a_{4} form an arithmetic sequence, their average equals the middle term, which gives us: [6a_{2} = a_{4} + a_{3}] Substituting a_{2} = a_{1}q and a_{3} = a_{1}q^{2}, and a_{4} = a_{1}q^{3} into the equation, we get: [6a_{1}q = a_{1}(q^{3} + q^{2})] Simplifying, we find that a_{1} = 1 and q = 2. Therefore, the general formula for the sequence {a_{n}} is: [a_{n} = 2^{n-1}] Part 2: Finding the Sum of the First n Terms of Sequence {b_{n}} Given b_{n} = a_{n}log_{2}a_{n+1}^{2}, and from part (1), a_{n} = 2^{n-1}, we substitute to find: [b_{n} = 2^{n-1} log_{2}(2^{n})^{2} = 2^{n-1} cdot 2n = n cdot 2^{n}] To find the sum T_{n} of the first n terms of {b_{n}}, we consider: [T_{n} = 2 + 2 cdot 2^{2} + 3 cdot 2^{3} + ldots + n cdot 2^{n}] Doubling T_{n} and shifting the terms, we get: [2T_{n} = 2^{2} + 2 cdot 2^{3} + ldots + (n-1) cdot 2^{n} + n cdot 2^{n+1}] Subtracting the original series from this new series, we find: [-T_{n} = 2 + 2^{2} + 2^{3} + ldots + 2^{n} - n cdot 2^{n+1}] This simplifies to: [T_{n} = left(n-1right) cdot 2^{n+1} + 2] Therefore, the sum of the first n terms of the sequence {b_{n}} is: [boxed{T_{n} = left(n-1right) cdot 2^{n+1} + 2}]

answer:1. **Initial Price Assumption** Let the initial price of the gas be ( x ) forints per cubic meter. 2. **Price Increase Calculations** The first price increase is by 5%, so the new price becomes: [ x times 1.05 = 1.05x ] The second price increase is by 6%, so the new price after the second increase is: [ 1.05x times 1.06 = 1.113x ] The third price increase is by 10%, so the new price after the third increase is: [ 1.113x times 1.10 = 1.2243x ] 3. **Price Increase Prediction** Analysts predict that the total increase for the year will be by ( frac{1}{3} ) of the initial price, implying the final price will be: [ frac{4}{3}x ] 4. **Additional Percentage Increase Required** Let the additional percentage increase required be ( p % ). Then the final price incorporating this increase can be written as: [ 1.2243x times left(1 + frac{p}{100}right) = frac{4}{3}x ] 5. **Setting Up the Equation** Simplify the equation by canceling ( x ) from both sides: [ 1.2243 left(1 + frac{p}{100}right) = frac{4}{3} ] 6. **Solving for ( p )** [ 1 + frac{p}{100} = frac{frac{4}{3}}{1.2243} ] Calculate the right-hand side: [ frac{4}{3} = 1.overline{3}, quad frac{1.overline{3}}{1.2243} approx 1.3057 ] Setting up the equation: [ 1 + frac{p}{100} = 1.3057 ] Extract ( frac{p}{100} ): [ frac{p}{100} = 1.3057 - 1 ] Simplify: [ frac{p}{100} = 0.3057 ] Solve for ( p ): [ p = 0.3057 times 100 = 30.57 ] This does not seem right, let’s recheck the calculation: [ frac{4}{3 times 1.2243} approx 1.09, Rightarrow p = 0.09 times 100 Rightarrow p = 9 ] 7. **Verifying Results** The previous steps hold; let's simplify the verification quickly: [ 1.2243 left(1 + frac{p}{100}right) = frac{4}{3} Rightarrow 1.2243 + 0.12243 approx 1.overline{333} ] 8. **Conclusion:** Thus, the additional percentage increase required is approximately 8.91%. [ boxed{8.91%} ]

answer:To find out how long the shorter meeting is in minutes, we need to first understand the ratio of planning time to meeting time for the long hour meeting. For the long hour meeting: Planning time = 2 hours Paperwork preparation time = 7 hours Total preparation time = 2 hours + 7 hours = 9 hours We don't know the actual length of the long hour meeting, but we know the total preparation time is 9 hours. For the shorter meeting: Total preparation time = 4.5 hours Since the preparation time for the shorter meeting is 4.5 hours, and we assume the ratio of preparation time to meeting time is the same for both meetings, we can set up a proportion to find the length of the shorter meeting. Let's call the length of the shorter meeting in hours "x". The ratio of preparation time to meeting time for the long hour meeting should be equal to the ratio for the shorter meeting: 9 hours preparation / long meeting time = 4.5 hours preparation / x hours meeting time Since we don't have the actual length of the long meeting, we cannot solve for x directly. However, the question seems to be asking for the length of the shorter meeting in minutes, given that they need 4.5 hours for preparation. If we assume that the preparation time is directly proportional to the meeting time, then the length of the shorter meeting would be 4.5 hours. To convert 4.5 hours to minutes, we use the conversion factor that 1 hour = 60 minutes. 4.5 hours * 60 minutes/hour = 270 minutes Therefore, the shorter meeting is boxed{270} minutes long.