answer：First, determine the total number of people at the gathering. Alex invited 8 friends, and each of those friends brought 2 additional people. This means there are 8 friends plus 8 times 2 additional people, totaling 24 people.Next, calculate how many groups of 5 people can be formed from the 24 attendees. Dividing 24 by 5 gives 4.8, but since you can't have a fraction of a group, you round down to the nearest whole number, which is 4 groups.Therefore, 4 groups were formed at the gathering.

answer：First, I need to determine the total time Alex will spend on contacting literary agents and publishers.For the literary agents:- There are 15 agents.- Each email takes 20 minutes.- Total time for agents = 15 agents * 20 minutes = 300 minutes.For the publishers:- There are 10 publishers.- Each email takes 30 minutes.- Total time for publishers = 10 publishers * 30 minutes = 300 minutes.Adding both, the total time required is 300 + 300 = 600 minutes.Next, I need to convert the daily available time from hours to minutes:- Alex dedicates 5 hours each day.- 5 hours * 60 minutes/hour = 300 minutes per day.Finally, to find out how many days it will take:- Total time needed / Daily available time = 600 minutes / 300 minutes/day = 2 days.

answer：Alright, so I've got this problem about a novelist who writes pages following a sequence similar to Fibonacci but with a twist. The sequence is defined by ( a_n = a_{n-1} + a_{n-2} + 1 ) with initial terms ( a_1 = 1 ) and ( a_2 = 1 ). I need to find an explicit formula for the ( n )-th term and then determine the smallest ( n > 2 ) such that ( a_n ) is a prime number.First, let me try to understand the sequence. It's similar to Fibonacci, which is ( F_n = F_{n-1} + F_{n-2} ), but here we have an extra "+1" each time. So each term is the sum of the two previous terms plus one. Let me write out the first few terms to see the pattern.Given ( a_1 = 1 ) and ( a_2 = 1 ), let's compute ( a_3 ) to ( a_6 ):- ( a_3 = a_2 + a_1 + 1 = 1 + 1 + 1 = 3 )- ( a_4 = a_3 + a_2 + 1 = 3 + 1 + 1 = 5 )- ( a_5 = a_4 + a_3 + 1 = 5 + 3 + 1 = 9 )- ( a_6 = a_5 + a_4 + 1 = 9 + 5 + 1 = 15 )- ( a_7 = a_6 + a_5 + 1 = 15 + 9 + 1 = 25 )- ( a_8 = a_7 + a_6 + 1 = 25 + 15 + 1 = 41 )- ( a_9 = a_8 + a_7 + 1 = 41 + 25 + 1 = 67 )- ( a_{10} = a_9 + a_8 + 1 = 67 + 41 + 1 = 109 )Hmm, interesting. So the sequence goes: 1, 1, 3, 5, 9, 15, 25, 41, 67, 109, ...I notice that starting from ( a_3 ), each term is the sum of the two previous terms plus one. So, this is a linear recurrence relation with constant coefficients. The general form of such a recurrence is ( a_n - a_{n-1} - a_{n-2} = 1 ). To solve this, I can use methods for solving linear recurrence relations.First, let me write the homogeneous part of the recurrence: ( a_n - a_{n-1} - a_{n-2} = 0 ). The characteristic equation for this is ( r^2 - r - 1 = 0 ). Solving this quadratic equation:( r = [1 pm sqrt{1 + 4}]/2 = [1 pm sqrt{5}]/2 ).So the roots are ( r_1 = (1 + sqrt{5})/2 ) and ( r_2 = (1 - sqrt{5})/2 ). These are the golden ratio and its conjugate.Therefore, the general solution to the homogeneous equation is:( a_n^{(h)} = C_1 r_1^n + C_2 r_2^n ).But our original equation is nonhomogeneous because of the "+1" term. So we need a particular solution ( a_n^{(p)} ).Since the nonhomogeneous term is a constant (1), we can try a constant particular solution. Let me assume ( a_n^{(p)} = K ), where K is a constant.Plugging into the recurrence:( K - K - K = 1 ) => ( -K = 1 ) => ( K = -1 ).So the general solution is:( a_n = a_n^{(h)} + a_n^{(p)} = C_1 r_1^n + C_2 r_2^n - 1 ).Now, we need to determine the constants ( C_1 ) and ( C_2 ) using the initial conditions.Given ( a_1 = 1 ) and ( a_2 = 1 ), let's set up the equations.For ( n = 1 ):( a_1 = C_1 r_1 + C_2 r_2 - 1 = 1 ).So,( C_1 r_1 + C_2 r_2 = 2 ). (Equation 1)For ( n = 2 ):( a_2 = C_1 r_1^2 + C_2 r_2^2 - 1 = 1 ).So,( C_1 r_1^2 + C_2 r_2^2 = 2 ). (Equation 2)Now, I need to compute ( r_1^2 ) and ( r_2^2 ).Recall that ( r_1 = (1 + sqrt{5})/2 ) and ( r_2 = (1 - sqrt{5})/2 ).Compute ( r_1^2 ):( r_1^2 = [(1 + sqrt{5})/2]^2 = (1 + 2sqrt{5} + 5)/4 = (6 + 2sqrt{5})/4 = (3 + sqrt{5})/2 ).Similarly, ( r_2^2 = [(1 - sqrt{5})/2]^2 = (1 - 2sqrt{5} + 5)/4 = (6 - 2sqrt{5})/4 = (3 - sqrt{5})/2 ).So, Equation 2 becomes:( C_1 (3 + sqrt{5})/2 + C_2 (3 - sqrt{5})/2 = 2 ).Let me write both equations:Equation 1: ( C_1 (1 + sqrt{5})/2 + C_2 (1 - sqrt{5})/2 = 2 ).Equation 2: ( C_1 (3 + sqrt{5})/2 + C_2 (3 - sqrt{5})/2 = 2 ).Let me denote ( A = C_1 ) and ( B = C_2 ) for simplicity.So,Equation 1: ( A (1 + sqrt{5})/2 + B (1 - sqrt{5})/2 = 2 ).Equation 2: ( A (3 + sqrt{5})/2 + B (3 - sqrt{5})/2 = 2 ).Let me multiply both equations by 2 to eliminate denominators:Equation 1: ( A (1 + sqrt{5}) + B (1 - sqrt{5}) = 4 ).Equation 2: ( A (3 + sqrt{5}) + B (3 - sqrt{5}) = 4 ).Now, let's write these as:Equation 1: ( A(1 + sqrt{5}) + B(1 - sqrt{5}) = 4 ).Equation 2: ( A(3 + sqrt{5}) + B(3 - sqrt{5}) = 4 ).Let me denote ( x = A ) and ( y = B ) for simplicity.So,1. ( x(1 + sqrt{5}) + y(1 - sqrt{5}) = 4 ).2. ( x(3 + sqrt{5}) + y(3 - sqrt{5}) = 4 ).Let me write this as a system of linear equations:Equation 1: ( (1 + sqrt{5})x + (1 - sqrt{5})y = 4 ).Equation 2: ( (3 + sqrt{5})x + (3 - sqrt{5})y = 4 ).Let me try to solve this system. Maybe I can use substitution or elimination. Let's try elimination.Let me denote ( S = x + y ) and ( D = x - y ). Hmm, but not sure if that helps. Alternatively, maybe express one variable in terms of the other.From Equation 1:( (1 + sqrt{5})x + (1 - sqrt{5})y = 4 ).Let me solve for x:( (1 + sqrt{5})x = 4 - (1 - sqrt{5})y ).So,( x = [4 - (1 - sqrt{5})y] / (1 + sqrt{5}) ).Now, plug this into Equation 2:( (3 + sqrt{5}) [4 - (1 - sqrt{5})y] / (1 + sqrt{5}) + (3 - sqrt{5})y = 4 ).This looks messy, but let's try to compute it step by step.First, compute the numerator of the first term:( (3 + sqrt{5})[4 - (1 - sqrt{5})y] ).Let me expand this:= ( (3 + sqrt{5}) * 4 - (3 + sqrt{5})(1 - sqrt{5})y ).Compute each part:First part: ( 4*(3 + sqrt{5}) = 12 + 4sqrt{5} ).Second part: ( (3 + sqrt{5})(1 - sqrt{5}) ).Multiply these:= ( 3*1 + 3*(-sqrt{5}) + sqrt{5}*1 + sqrt{5}*(-sqrt{5}) )= ( 3 - 3sqrt{5} + sqrt{5} - 5 )= ( (3 - 5) + (-3sqrt{5} + sqrt{5}) )= ( -2 - 2sqrt{5} ).So, the second part is ( (-2 - 2sqrt{5})y ).Therefore, the numerator is:( 12 + 4sqrt{5} - (-2 - 2sqrt{5})y )= ( 12 + 4sqrt{5} + 2y + 2sqrt{5}y ).Now, the entire first term is:( [12 + 4sqrt{5} + 2y + 2sqrt{5}y] / (1 + sqrt{5}) ).So, Equation 2 becomes:( [12 + 4sqrt{5} + 2y + 2sqrt{5}y] / (1 + sqrt{5}) + (3 - sqrt{5})y = 4 ).Let me denote ( (3 - sqrt{5})y ) as is for now.To simplify the first term, let's rationalize the denominator by multiplying numerator and denominator by ( (1 - sqrt{5}) ):So,( [12 + 4sqrt{5} + 2y + 2sqrt{5}y] * (1 - sqrt{5}) / [ (1 + sqrt{5})(1 - sqrt{5}) ] ).The denominator becomes ( 1 - 5 = -4 ).So, the first term is:( [12 + 4sqrt{5} + 2y + 2sqrt{5}y] * (1 - sqrt{5}) / (-4) ).Let me compute the numerator:Multiply each term in the first bracket by ( (1 - sqrt{5}) ):= ( 12*(1 - sqrt{5}) + 4sqrt{5}*(1 - sqrt{5}) + 2y*(1 - sqrt{5}) + 2sqrt{5}y*(1 - sqrt{5}) ).Compute each part:1. ( 12*(1 - sqrt{5}) = 12 - 12sqrt{5} ).2. ( 4sqrt{5}*(1 - sqrt{5}) = 4sqrt{5} - 4*5 = 4sqrt{5} - 20 ).3. ( 2y*(1 - sqrt{5}) = 2y - 2ysqrt{5} ).4. ( 2sqrt{5}y*(1 - sqrt{5}) = 2sqrt{5}y - 2*5 y = 2sqrt{5}y - 10y ).Now, combine all these:= ( (12 - 12sqrt{5}) + (4sqrt{5} - 20) + (2y - 2ysqrt{5}) + (2sqrt{5}y - 10y) ).Combine like terms:Constants: 12 - 20 = -8.Terms with ( sqrt{5} ): -12sqrt{5} + 4sqrt{5} = (-8sqrt{5}).Terms with y: 2y - 10y = -8y.Terms with ( ysqrt{5} ): -2ysqrt{5} + 2ysqrt{5} = 0.So, total numerator is ( -8 - 8sqrt{5} - 8y ).Therefore, the first term is:( (-8 - 8sqrt{5} - 8y) / (-4) = (8 + 8sqrt{5} + 8y)/4 = 2 + 2sqrt{5} + 2y ).So, Equation 2 becomes:( 2 + 2sqrt{5} + 2y + (3 - sqrt{5})y = 4 ).Combine like terms:The constants: 2 + 2sqrt{5}.The y terms: 2y + (3 - sqrt{5})y = (2 + 3 - sqrt{5})y = (5 - sqrt{5})y.So,( 2 + 2sqrt{5} + (5 - sqrt{5})y = 4 ).Subtract constants:( (5 - sqrt{5})y = 4 - 2 - 2sqrt{5} = 2 - 2sqrt{5} ).Therefore,( y = (2 - 2sqrt{5}) / (5 - sqrt{5}) ).Simplify this fraction by rationalizing the denominator:Multiply numerator and denominator by ( 5 + sqrt{5} ):Numerator: ( (2 - 2sqrt{5})(5 + sqrt{5}) ).Denominator: ( (5 - sqrt{5})(5 + sqrt{5}) = 25 - 5 = 20 ).Compute numerator:= ( 2*5 + 2*sqrt{5} - 2sqrt{5}*5 - 2sqrt{5}*sqrt{5} )= ( 10 + 2sqrt{5} - 10sqrt{5} - 10 )= ( (10 - 10) + (2sqrt{5} - 10sqrt{5}) )= ( 0 - 8sqrt{5} )= ( -8sqrt{5} ).So, ( y = (-8sqrt{5}) / 20 = (-2sqrt{5}) / 5 ).So, ( y = -2sqrt{5}/5 ).Recall that ( y = B = C_2 ).Now, let's find x from Equation 1.From Equation 1:( (1 + sqrt{5})x + (1 - sqrt{5})y = 4 ).We have y = -2√5 /5.Plug in:( (1 + sqrt{5})x + (1 - sqrt{5})(-2sqrt{5}/5) = 4 ).Compute the second term:( (1 - sqrt{5})(-2sqrt{5}/5) = (-2sqrt{5}/5)(1 - sqrt{5}) ).= ( (-2sqrt{5}/5) + (2*5)/5 ).= ( (-2sqrt{5}/5) + 2 ).So, the equation becomes:( (1 + sqrt{5})x + (-2sqrt{5}/5 + 2) = 4 ).Bring constants to the right:( (1 + sqrt{5})x = 4 - (-2sqrt{5}/5 + 2) ).= ( 4 - 2 + 2sqrt{5}/5 ).= ( 2 + 2sqrt{5}/5 ).Factor out 2:= ( 2(1 + sqrt{5}/5) ).= ( 2( (5 + sqrt{5}) /5 ) ).= ( (10 + 2sqrt{5}) /5 ).So,( x = [ (10 + 2sqrt{5}) /5 ] / (1 + sqrt{5}) ).Simplify:= ( (10 + 2sqrt{5}) / [5(1 + sqrt{5})] ).Factor numerator:= ( 2(5 + sqrt{5}) / [5(1 + sqrt{5})] ).Notice that ( 5 + sqrt{5} = 5 + sqrt{5} ), and ( 1 + sqrt{5} ) is the same as before.So,= ( 2(5 + sqrt{5}) / [5(1 + sqrt{5})] ).We can factor numerator and denominator:Note that ( 5 + sqrt{5} = 5 + sqrt{5} ), and ( 1 + sqrt{5} ) is as is.Alternatively, rationalize the denominator:Multiply numerator and denominator by ( (1 - sqrt{5}) ):Numerator: ( 2(5 + sqrt{5})(1 - sqrt{5}) ).Denominator: ( 5(1 + sqrt{5})(1 - sqrt{5}) = 5(1 - 5) = 5*(-4) = -20 ).Compute numerator:= ( 2[5*(1) + 5*(-sqrt{5}) + sqrt{5}*1 + sqrt{5}*(-sqrt{5})] ).= ( 2[5 - 5sqrt{5} + sqrt{5} - 5] ).= ( 2[ (5 - 5) + (-5sqrt{5} + sqrt{5}) ] ).= ( 2[ 0 - 4sqrt{5} ] ).= ( 2*(-4sqrt{5}) = -8sqrt{5} ).So, numerator is ( -8sqrt{5} ), denominator is ( -20 ).Thus,( x = (-8sqrt{5}) / (-20) = (8sqrt{5}) / 20 = (2sqrt{5}) / 5 ).So, ( x = 2sqrt{5}/5 ).Therefore, ( C_1 = x = 2sqrt{5}/5 ) and ( C_2 = y = -2sqrt{5}/5 ).So, plugging back into the general solution:( a_n = C_1 r_1^n + C_2 r_2^n - 1 ).Substituting the values:( a_n = (2sqrt{5}/5) * [(1 + sqrt{5})/2]^n + (-2sqrt{5}/5) * [(1 - sqrt{5})/2]^n - 1 ).Simplify this expression.First, note that ( [(1 + sqrt{5})/2]^n ) is often denoted as ( phi^n ), where ( phi ) is the golden ratio. Similarly, ( [(1 - sqrt{5})/2]^n ) is ( psi^n ), where ( psi ) is the conjugate.So, the expression becomes:( a_n = (2sqrt{5}/5) phi^n + (-2sqrt{5}/5) psi^n - 1 ).Factor out ( 2sqrt{5}/5 ):= ( (2sqrt{5}/5)(phi^n - psi^n) - 1 ).But I recall that ( phi^n - psi^n ) is related to the Fibonacci sequence. Specifically, ( phi^n - psi^n = sqrt{5} F_n ), where ( F_n ) is the n-th Fibonacci number.Wait, let me verify that.Yes, indeed, the closed-form expression for Fibonacci numbers is Binet's formula:( F_n = ( phi^n - psi^n ) / sqrt{5} ).Therefore, ( phi^n - psi^n = sqrt{5} F_n ).So, substituting back into our expression:( a_n = (2sqrt{5}/5)(sqrt{5} F_n) - 1 ).Simplify:= ( (2*5 /5) F_n - 1 ).= ( 2 F_n - 1 ).Wow, that's a much simpler expression!So, the explicit formula is ( a_n = 2F_n - 1 ).Let me check this with the terms I computed earlier.Given ( a_1 = 1 ), ( F_1 = 1 ), so ( 2*1 -1 = 1 ). Correct.( a_2 = 1 ), ( F_2 = 1 ), so ( 2*1 -1 =1 ). Correct.( a_3 = 3 ), ( F_3 = 2 ), so ( 2*2 -1 =3 ). Correct.( a_4 =5 ), ( F_4 =3 ), so ( 2*3 -1=5 ). Correct.( a_5 =9 ), ( F_5=5 ), so ( 2*5 -1=9 ). Correct.( a_6=15 ), ( F_6=8 ), so ( 2*8 -1=15 ). Correct.( a_7=25 ), ( F_7=13 ), so ( 2*13 -1=25 ). Correct.Great, so the explicit formula is indeed ( a_n = 2F_n -1 ).Now, moving on to the second part: find the smallest ( n > 2 ) such that ( a_n ) is a prime number.Given that ( a_n = 2F_n -1 ), we need to find the smallest ( n > 2 ) where ( 2F_n -1 ) is prime.From the earlier terms:- ( a_3 = 3 ), which is prime.Wait, but n=3 is greater than 2, so is 3 the answer? Wait, n=3 is the first term after n=2, so is 3 prime? Yes, 3 is prime.But let me check the problem statement: it says "the smallest ( n > 2 )", so n=3 is the first candidate.But let me confirm if ( a_3 ) is indeed 3.Yes, ( a_3 = a_2 + a_1 +1 =1 +1 +1=3 ). So, 3 is prime.Wait, but hold on, n=3 is the third term, which is greater than 2, so n=3 is the answer.But wait, let me check the problem statement again: "the smallest ( n > 2 ) such that ( a_n ) is a prime number."So, n=3 is the first n>2, and ( a_3=3 ) is prime. So, n=3 is the answer.But just to be thorough, let's check the next few terms:- ( a_4=5 ), which is prime.- ( a_5=9 ), which is not prime.- ( a_6=15 ), not prime.- ( a_7=25 ), not prime.- ( a_8=41 ), prime.- ( a_9=67 ), prime.- ( a_{10}=109 ), prime.So, the primes occur at n=3,4,8,9,10,...But the question is asking for the smallest ( n > 2 ), which is n=3.Wait, but hold on, n=3 is the third term, which is greater than 2, so yes, n=3 is the answer.But just to make sure, let's compute ( a_3 ) again.Given ( a_1=1 ), ( a_2=1 ), so ( a_3 =1+1+1=3 ). Correct.So, 3 is prime, so n=3 is the answer.But wait, the problem says "the smallest ( n > 2 )", so n=3 is indeed the answer.However, I want to make sure that the explicit formula is correct because sometimes when dealing with recurrence relations, especially with Fibonacci-like sequences, the indexing can be tricky.Wait, in the Fibonacci sequence, ( F_1 =1 ), ( F_2=1 ), ( F_3=2 ), ( F_4=3 ), etc. So, in our case, ( a_n = 2F_n -1 ).So, for n=1: ( a_1=2*1 -1=1 ).n=2: ( a_2=2*1 -1=1 ).n=3: ( a_3=2*2 -1=3 ).n=4: ( a_4=2*3 -1=5 ).n=5: ( a_5=2*5 -1=9 ).Yes, that's correct.So, the first prime after n=2 is at n=3.Therefore, the answer is n=3.But just to be absolutely sure, let me check if the problem counts n=3 as greater than 2. Yes, 3>2, so it's acceptable.Thus, the explicit formula is ( a_n = 2F_n -1 ), and the smallest ( n > 2 ) with ( a_n ) prime is n=3.

answer：First, I need to determine the number of infections remaining after one year using both traditional methods and Jamie's biotechnology method.Starting with the initial number of infections, which is 4,000.For the traditional method, it reduces infections by 30%. So, I'll calculate 30% of 4,000 and subtract that from the initial number to find the remaining infections.Next, for Jamie's biotechnology method, it reduces infections by 50%. I'll calculate 50% of 4,000 and subtract that from the initial number to find the remaining infections using this method.Finally, to find out how many more infections are reduced using the biotechnology method compared to the traditional method, I'll subtract the number of infections remaining with the biotechnology method from the number remaining with the traditional method.

answer：First, I need to calculate the total calories John burns from walking each morning. He walks 3 miles each day and burns 100 calories per mile. So, for one day, that's 3 miles multiplied by 100 calories per mile, which equals 300 calories burned from walking each day.Next, I'll calculate the calories burned from swimming each evening. John swims 20 laps each day and burns 15 calories per lap. For one day, that's 20 laps multiplied by 15 calories per lap, totaling 300 calories burned from swimming each day.Now, to find the total calories burned each day from both activities, I'll add the calories burned from walking and swimming. That's 300 calories from walking plus 300 calories from swimming, which equals 600 calories burned per day.Finally, to determine the total calories burned over 5 days, I'll multiply the daily total by 5. So, 600 calories per day multiplied by 5 days equals 3,000 calories burned in total.

answer：First, I need to understand how the disease spreads in each town. In Town A, each infected person infects 3 more people every week, and in Town B, each infected person infects 2 more people every week.Next, I'll calculate the number of infected people in each town after 2 weeks. For Town A, starting with 5 infected people, the number of new infections each week is 5 multiplied by 3, which equals 15 new infections. After 2 weeks, the total infected people in Town A will be 5 plus 15, totaling 20.For Town B, starting with 8 infected people, the number of new infections each week is 8 multiplied by 2, which equals 16 new infections. After 2 weeks, the total infected people in Town B will be 8 plus 16, totaling 24.Finally, to find the total number of infected people across both towns after 2 weeks, I'll add the totals from Town A and Town B: 20 plus 24 equals 44.