🧠:<think>To determine how many hours Alex can afford with the lawyer's discount, I'll start by setting up an inequality. The lawyer charges 150 per hour with a 10% discount for over 100 hours. The total cost after the discount should not exceed 14,000.First, I'll calculate the discounted hourly rate, which is 90% of 150, or 135 per hour. Then, I'll set up the inequality 135h ≤ 14,000 and solve for h. This gives me the maximum number of hours Alex can afford, which is approximately 103.7 hours.Next, to calculate the total time Alex spent with their children in a non-leap year, I'll consider the number of weekdays and weekends. There are 52 weeks in a year, with 5 weekdays and 2 weekend days each week. Alex spends 2 hours on weekdays and 5 hours on weekends. Multiplying these by the number of weekdays and weekends respectively gives the total hours spent with the children.For the goal of a 10% increase in time spent next year, I'll first determine the current total hours and then calculate 10% of that. Dividing the additional hours needed by the number of weeks in a year will give the extra hours Alex needs to allocate each week.</think>

🧠:<think>First, I need to determine the total number of eligible voters in Bolton. I know that the voter turnout was 75% of the eligible voters, and the total votes cast were 36,000. To find the eligible voters, I can set up the equation where 75% of the eligible voters equals 36,000. By dividing 36,000 by 0.75, I can calculate the total number of eligible voters.Next, for the second part, I need to calculate the new total number of votes the Conservative Party would receive if there's a 5% increase in voter turnout, while keeping their vote percentage constant at 55%. First, I'll find the new voter turnout percentage by adding 5% to the original 75%, resulting in an 80% turnout. Then, I'll calculate the new total votes cast by taking 80% of the eligible voters. Finally, I'll determine the Conservative Party's votes by taking 55% of this new total.</think>

🧠:<think>First, I need to calculate the flight time from Amsterdam to Casablanca. The plane's average speed is 850 km/h, but there's a headwind that reduces its effective speed by 10%. So, the effective speed is 850 km/h multiplied by 0.9, which equals 765 km/h. The flight distance is 2,300 km, so the flight time is 2,300 km divided by 765 km/h, resulting in approximately 3.0065 hours.Next, I should add the 1-hour layover in Casablanca. This brings the total time for the flight and layover to about 4.0065 hours, which rounds to 4.01 hours.For the train journey from Casablanca to Rabat, the total distance is 95 km. The train normally travels at 150 km/h, but it must reduce its speed by 20% for 30 km of the journey. The reduced speed is 150 km/h multiplied by 0.8, which equals 120 km/h. I'll calculate the time for the reduced speed section: 30 km divided by 120 km/h equals 0.25 hours. For the remaining 65 km, the train travels at the normal speed of 150 km/h, so the time is 65 km divided by 150 km/h, which is approximately 0.4333 hours.Adding these two times together gives a total train travel time of about 0.6833 hours, which rounds to 0.68 hours.</think>

🧠:<think>Okay, so I have this problem here about a Romanian immigrant who's into literature and now exploring math sequences. The first part is about a sequence inspired by a traditional tale, which sounds like the wheat and chessboard problem. The sequence is given as ( a_n = 2^{n-1} ) for ( n geq 1 ). I need to find the closed form of the sum ( S = sum_{n=1}^{k} a_n ) and determine the smallest integer ( k ) such that ( S > 1,000,000 ).Alright, let me start by recalling what a geometric series is. A geometric series is a series where each term is a constant multiple of the previous term. In this case, the sequence ( a_n = 2^{n-1} ) is a geometric sequence with the first term ( a_1 = 2^{0} = 1 ) and the common ratio ( r = 2 ). So, the sum ( S ) is a geometric series.The formula for the sum of the first ( k ) terms of a geometric series is ( S = a_1 times frac{r^k - 1}{r - 1} ). Plugging in the values we have, ( a_1 = 1 ) and ( r = 2 ), so the sum becomes:( S = frac{2^k - 1}{2 - 1} = 2^k - 1 ).So, the closed form of the sum is ( S = 2^k - 1 ). Now, we need to find the smallest integer ( k ) such that ( S > 1,000,000 ). That translates to:( 2^k - 1 > 1,000,000 ).Adding 1 to both sides:( 2^k > 1,000,001 ).Now, I need to find the smallest integer ( k ) where this inequality holds. To solve for ( k ), I can take the logarithm base 2 of both sides:( k > log_2(1,000,001) ).I remember that ( log_2(1,048,576) = 20 ) because ( 2^{20} = 1,048,576 ). Since 1,000,001 is less than 1,048,576, ( log_2(1,000,001) ) must be slightly less than 20. Let me compute it more precisely.Using the change of base formula, ( log_2(x) = frac{ln x}{ln 2} ). Let's compute ( ln(1,000,001) ) and ( ln 2 ).First, ( ln(1,000,001) ). I know that ( ln(1,000,000) = ln(10^6) = 6 ln(10) approx 6 times 2.302585 = 13.81551 ). Since 1,000,001 is just 1 more than 1,000,000, the natural log won't change much. Maybe approximately 13.81551 plus a tiny bit. Let's say approximately 13.81551 + 0.000001 ≈ 13.815511.Then, ( ln 2 approx 0.693147 ).So, ( log_2(1,000,001) approx frac{13.815511}{0.693147} approx 19.931 ).Therefore, ( k > 19.931 ). Since ( k ) must be an integer, the smallest integer greater than 19.931 is 20. Let me verify:Compute ( 2^{20} = 1,048,576 ). Then, ( S = 1,048,576 - 1 = 1,048,575 ). Is 1,048,575 greater than 1,000,000? Yes, it is. What about ( k = 19 )?Compute ( 2^{19} = 524,288 ). Then, ( S = 524,288 - 1 = 524,287 ). That's less than 1,000,000. So, indeed, the smallest integer ( k ) is 20.So, the closed form is ( S = 2^k - 1 ), and the smallest ( k ) is 20.Moving on to the second problem. The individual defines a modified Fibonacci sequence ( F_n ) where ( F_0 = 1 ), ( F_1 = 1 ), and for ( n geq 2 ), ( F_n = F_{n-1} + F_{n-2} + 1 ). I need to prove that the ratio ( frac{F_{n+1}}{F_n} ) converges as ( n to infty ) and find its limit, relating it to the golden ratio.Hmm, okay. So, this is a variation of the Fibonacci sequence where each term is the sum of the two previous terms plus 1. The standard Fibonacci sequence has a ratio that converges to the golden ratio ( phi = frac{1 + sqrt{5}}{2} approx 1.618 ). But here, we have an extra 1 added each time. I wonder how that affects the ratio.First, let me write down the recurrence relation:( F_n = F_{n-1} + F_{n-2} + 1 ).I need to analyze the behavior of ( frac{F_{n+1}}{F_n} ) as ( n ) becomes large. Let me denote ( r_n = frac{F_{n+1}}{F_n} ). Then, as ( n ) approaches infinity, if the limit exists, let's say ( r = lim_{n to infty} r_n ).Assuming the limit exists, we can write:( r = lim_{n to infty} frac{F_{n+1}}{F_n} = lim_{n to infty} frac{F_n + F_{n-1} + 1}{F_n} ).Wait, let me substitute the recurrence relation into ( F_{n+1} ):( F_{n+1} = F_n + F_{n-1} + 1 ).So, ( r_n = frac{F_{n+1}}{F_n} = frac{F_n + F_{n-1} + 1}{F_n} = 1 + frac{F_{n-1}}{F_n} + frac{1}{F_n} ).Let me denote ( r_{n-1} = frac{F_n}{F_{n-1}} ), so ( frac{F_{n-1}}{F_n} = frac{1}{r_{n-1}} ). Then, the expression becomes:( r_n = 1 + frac{1}{r_{n-1}} + frac{1}{F_n} ).As ( n ) becomes large, ( F_n ) grows exponentially (since it's similar to Fibonacci but with an extra term). So, ( frac{1}{F_n} ) tends to 0. Therefore, in the limit as ( n to infty ), we have:( r = 1 + frac{1}{r} ).So, the equation becomes:( r = 1 + frac{1}{r} ).Multiplying both sides by ( r ):( r^2 = r + 1 ).Bringing all terms to one side:( r^2 - r - 1 = 0 ).This is the same quadratic equation as the golden ratio. Solving it:( r = frac{1 pm sqrt{1 + 4}}{2} = frac{1 pm sqrt{5}}{2} ).Since the ratio ( r ) must be positive, we discard the negative root:( r = frac{1 + sqrt{5}}{2} approx 1.618 ).So, the limit is the golden ratio. That's interesting because even though we added an extra 1 each time, the ratio still converges to the golden ratio. That might have a poetic interpretation, perhaps symbolizing that even with additional elements, the fundamental beauty and proportion remain unchanged, much like how adding elements in art or literature can enhance the work without altering its core essence.But wait, let me make sure I didn't skip any steps. I assumed that ( lim_{n to infty} r_n = r ) exists. I need to confirm that the sequence ( r_n ) actually converges.Looking at the recurrence relation, as ( n ) increases, the term ( frac{1}{F_n} ) becomes negligible, so the behavior is dominated by ( 1 + frac{1}{r_{n-1}} ). If the sequence ( r_n ) approaches a limit, then the equation ( r = 1 + frac{1}{r} ) holds, which we solved.To check convergence, perhaps I can analyze the behavior of ( r_n ). Let's compute the first few terms to see if it seems to approach the golden ratio.Given ( F_0 = 1 ), ( F_1 = 1 ).Compute ( F_2 = F_1 + F_0 + 1 = 1 + 1 + 1 = 3 ).( F_3 = F_2 + F_1 + 1 = 3 + 1 + 1 = 5 ).( F_4 = F_3 + F_2 + 1 = 5 + 3 + 1 = 9 ).( F_5 = 9 + 5 + 1 = 15 ).( F_6 = 15 + 9 + 1 = 25 ).( F_7 = 25 + 15 + 1 = 41 ).( F_8 = 41 + 25 + 1 = 67 ).( F_9 = 67 + 41 + 1 = 109 ).( F_{10} = 109 + 67 + 1 = 177 ).Now, compute the ratios:( r_1 = F_2 / F_1 = 3 / 1 = 3 ).( r_2 = F_3 / F_2 = 5 / 3 ≈ 1.6667 ).( r_3 = F_4 / F_3 = 9 / 5 = 1.8 ).( r_4 = F_5 / F_4 = 15 / 9 ≈ 1.6667 ).( r_5 = F_6 / F_5 = 25 / 15 ≈ 1.6667 ).( r_6 = F_7 / F_6 = 41 / 25 ≈ 1.64 ).( r_7 = F_8 / F_7 = 67 / 41 ≈ 1.634 ).( r_8 = F_9 / F_8 = 109 / 67 ≈ 1.6269 ).( r_9 = F_{10} / F_9 = 177 / 109 ≈ 1.6239 ).Hmm, the ratios are oscillating but seem to be converging towards approximately 1.618, which is the golden ratio. So, this numerical evidence supports the analytical result.Therefore, the ratio ( frac{F_{n+1}}{F_n} ) does converge to the golden ratio ( phi = frac{1 + sqrt{5}}{2} ) as ( n to infty ).This poetic interpretation could be that despite the additional term (the +1 in the recurrence), the sequence still approaches the same proportion as the classic Fibonacci sequence, symbolizing that even with modifications, the essence or beauty remains intact, much like how adding elements in a poem or story can enhance it without losing its core theme.So, to summarize:1. The sum ( S = 2^k - 1 ), and the smallest ( k ) is 20.2. The ratio converges to the golden ratio ( phi = frac{1 + sqrt{5}}{2} ).Final Answer1. The closed form of the sum is ( boxed{2^k - 1} ) and the smallest integer ( k ) is ( boxed{20} ).2. The limit of the ratio is ( boxed{frac{1 + sqrt{5}}{2}} ).</think>