answer：First, I need to determine the number of calories each meal and each snack should provide based on the client's total daily caloric intake of 2,400 calories.The meal plan consists of 3 meals and 2 snacks. Each meal is supposed to provide 35% of the total calories, and each snack should provide 7.5% of the total calories.To find the calories per meal, I'll calculate 35% of 2,400 calories. Similarly, for each snack, I'll calculate 7.5% of 2,400 calories.After performing these calculations, I'll present the results clearly to ensure the meal plan meets the client's needs for high-intensity interval training.

answer：First, I need to determine the amount spent on digital marketing in each country before the increases.For the first country, the spending increased by 15% to reach 115 million. To find the original amount, I'll divide 115 million by 1.15.In the second country, the spending increased by 20% to reach 180 million. I'll divide 180 million by 1.20 to find the original amount.For the third country, the spending increased by 10% to reach 88 million. I'll divide 88 million by 1.10 to find the original amount.After calculating the original amounts for each country, I'll sum them up to find the total amount spent on digital marketing across all three countries before the increases.

answer：First, I need to determine how many DVDs Alex has in each genre. He has a total of 120 Warner Bros. DVDs.25% of his collection are action movies. To find the number of action DVDs:25% of 120 = 0.25 × 120 = 30 DVDs.30% of his collection are comedies. To find the number of comedy DVDs:30% of 120 = 0.30 × 120 = 36 DVDs.The remaining DVDs are classic dramas. To find the number of drama DVDs:120 - 30 (action) - 36 (comedy) = 54 DVDs.Next, I need to calculate how many shelves Alex needs for each genre, given that each shelf holds exactly 10 DVDs.For action movies:30 DVDs ÷ 10 per shelf = 3 shelves.For comedy movies:36 DVDs ÷ 10 per shelf = 3.6 shelves. Since Alex can't have a fraction of a shelf, he'll need 4 shelves.For classic dramas:54 DVDs ÷ 10 per shelf = 5.4 shelves. Again, rounding up, he'll need 6 shelves.Finally, to find the total number of shelves needed:3 (action) + 4 (comedy) + 6 (drama) = 13 shelves.

answer：First, I need to calculate Clara's earnings from each of the three contracts separately.For the first contract, the property is worth 450,000 and Clara charges 1.5% of the property value. To find her fee, I'll multiply 450,000 by 0.015.Next, for the second contract, the property is valued at 300,000 with a fee of 2%. I'll calculate this by multiplying 300,000 by 0.02.The third contract has a property worth 750,000, and Clara charges a flat fee of 5,000. Since this is a fixed amount, there's no need for additional calculations here.Finally, I'll add up the earnings from all three contracts to determine Clara's total earnings for the week.

answer：首先，我需要计算这位国学爱好者总共拥有多少本古籍。他有20本《论语》，15本《孟子》，以及12本《大学》。将这些数量相加，20加15等于35，再加上12，总共是47本古籍。接下来，他计划将这些书平均分给他的3个朋友。因此，我需要将总数量47除以3。47除以3等于15余2。这意味着每个朋友可以得到15本古籍，而还剩下2本无法平均分配。最后，我需要将结果以清晰的方式呈现，说明每个朋友得到的具体数量以及剩余的情况。

answer：Okay, so I have this problem about the electoral performance of the Papandreou family in Greece. The number of votes they received in each election cycle follows a recursion relation. Let me try to figure this out step by step.First, the problem states that the number of votes ( V_i ) in election ( i ) is defined by the recursion:[ V_{i+2} = 2V_{i+1} + 3V_i ]with initial conditions ( V_0 = 1000 ) and ( V_1 = 1500 ).I need to find a closed-form expression for ( V_i ). Hmm, this looks like a linear recurrence relation. I remember that for such recursions, we can solve them by finding the characteristic equation. Let me recall how that works.The general form of a linear recurrence relation is:[ V_{n+k} = a_{k-1}V_{n+k-1} + dots + a_0 V_n ]In this case, our recurrence is:[ V_{i+2} = 2V_{i+1} + 3V_i ]So, it's a second-order linear recurrence relation. The characteristic equation for this would be:[ r^2 = 2r + 3 ]Let me write that down:[ r^2 - 2r - 3 = 0 ]Now, I need to solve this quadratic equation for ( r ). Using the quadratic formula:[ r = frac{2 pm sqrt{(2)^2 - 4(1)(-3)}}{2(1)} ][ r = frac{2 pm sqrt{4 + 12}}{2} ][ r = frac{2 pm sqrt{16}}{2} ][ r = frac{2 pm 4}{2} ]So, the roots are:1. ( r = frac{2 + 4}{2} = frac{6}{2} = 3 )2. ( r = frac{2 - 4}{2} = frac{-2}{2} = -1 )Alright, so the roots are 3 and -1. Since these are distinct real roots, the general solution to the recurrence relation is:[ V_i = A cdot (3)^i + B cdot (-1)^i ]Where ( A ) and ( B ) are constants determined by the initial conditions.Now, I need to use the initial conditions to solve for ( A ) and ( B ).Given:- ( V_0 = 1000 )- ( V_1 = 1500 )Let's plug ( i = 0 ) into the general solution:[ V_0 = A cdot 3^0 + B cdot (-1)^0 ][ 1000 = A cdot 1 + B cdot 1 ][ 1000 = A + B ][ A + B = 1000 ] ...(1)Now, plug ( i = 1 ) into the general solution:[ V_1 = A cdot 3^1 + B cdot (-1)^1 ][ 1500 = 3A - B ][ 3A - B = 1500 ] ...(2)Now, I have a system of two equations:1. ( A + B = 1000 )2. ( 3A - B = 1500 )I can solve this system by adding the two equations together to eliminate ( B ):Adding equation (1) and equation (2):( (A + B) + (3A - B) = 1000 + 1500 )( A + B + 3A - B = 2500 )( 4A = 2500 )( A = frac{2500}{4} )( A = 625 )Now, substitute ( A = 625 ) back into equation (1):( 625 + B = 1000 )( B = 1000 - 625 )( B = 375 )So, the constants are ( A = 625 ) and ( B = 375 ). Therefore, the closed-form expression for ( V_i ) is:[ V_i = 625 cdot 3^i + 375 cdot (-1)^i ]Let me double-check this solution with the initial conditions to make sure I didn't make a mistake.For ( i = 0 ):[ V_0 = 625 cdot 3^0 + 375 cdot (-1)^0 = 625 + 375 = 1000 ] ✔️For ( i = 1 ):[ V_1 = 625 cdot 3^1 + 375 cdot (-1)^1 = 1875 - 375 = 1500 ] ✔️Good, that matches. Let me also check for ( i = 2 ) using the recursion to see if it holds.From the recursion:[ V_2 = 2V_1 + 3V_0 = 2*1500 + 3*1000 = 3000 + 3000 = 6000 ]Using the closed-form:[ V_2 = 625*3^2 + 375*(-1)^2 = 625*9 + 375*1 = 5625 + 375 = 6000 ] ✔️Perfect, that works too. So, I feel confident about the closed-form expression.Now, moving on to part 2: calculating ( V_{10} ) using this closed-form.So, ( V_{10} = 625 cdot 3^{10} + 375 cdot (-1)^{10} )First, compute ( 3^{10} ). Let me calculate that step by step:- ( 3^1 = 3 )- ( 3^2 = 9 )- ( 3^3 = 27 )- ( 3^4 = 81 )- ( 3^5 = 243 )- ( 3^6 = 729 )- ( 3^7 = 2187 )- ( 3^8 = 6561 )- ( 3^9 = 19683 )- ( 3^{10} = 59049 )So, ( 3^{10} = 59049 )Next, ( (-1)^{10} = 1 ) since any even power of -1 is 1.So, plugging these into the expression:[ V_{10} = 625 cdot 59049 + 375 cdot 1 ]Compute each term:First term: ( 625 * 59049 )Let me compute that. 625 is 5^4, which is 5*5*5*5. 59049 is 3^10, which is 59049.Alternatively, 625 * 59049 can be computed as:625 * 59049 = (600 + 25) * 59049 = 600*59049 + 25*59049Compute 600*59049:600 * 59049 = 600 * 59049Well, 600 is 6*100, so:6 * 59049 = 354,294Then, 354,294 * 100 = 35,429,400Now, 25 * 59049:25 * 59049 = (20 + 5) * 59049 = 20*59049 + 5*5904920*59049 = 1,180,9805*59049 = 295,245Add them together: 1,180,980 + 295,245 = 1,476,225Now, add the two parts together:35,429,400 + 1,476,225 = 36,905,625So, the first term is 36,905,625.Second term: 375 * 1 = 375.Therefore, ( V_{10} = 36,905,625 + 375 = 36,906,000 )Wait, let me verify that addition:36,905,625 + 375:36,905,625 + 300 = 36,905,92536,905,925 + 75 = 36,906,000Yes, that's correct.So, ( V_{10} = 36,906,000 ) votes.Let me just cross-verify this with the recursion to make sure I didn't make a mistake in the calculation.Alternatively, I can compute ( V_2 ) through ( V_{10} ) step by step using the recursion to see if I reach the same number.But that might take a while, but let me try a few steps to see if it's consistent.We already know:- ( V_0 = 1000 )- ( V_1 = 1500 )- ( V_2 = 6000 )Compute ( V_3 ):[ V_3 = 2V_2 + 3V_1 = 2*6000 + 3*1500 = 12,000 + 4,500 = 16,500 ]Using the closed-form:[ V_3 = 625*3^3 + 375*(-1)^3 = 625*27 + 375*(-1) = 16,875 - 375 = 16,500 ] ✔️Good.Compute ( V_4 ):[ V_4 = 2V_3 + 3V_2 = 2*16,500 + 3*6000 = 33,000 + 18,000 = 51,000 ]Closed-form:[ V_4 = 625*81 + 375*1 = 50,625 + 375 = 51,000 ] ✔️Good.Compute ( V_5 ):[ V_5 = 2V_4 + 3V_3 = 2*51,000 + 3*16,500 = 102,000 + 49,500 = 151,500 ]Closed-form:[ V_5 = 625*243 + 375*(-1)^5 = 625*243 + 375*(-1) ]Compute 625*243:243 * 600 = 145,800243 * 25 = 6,075So, 145,800 + 6,075 = 151,875Then, subtract 375: 151,875 - 375 = 151,500 ✔️Perfect.Compute ( V_6 ):[ V_6 = 2V_5 + 3V_4 = 2*151,500 + 3*51,000 = 303,000 + 153,000 = 456,000 ]Closed-form:[ V_6 = 625*729 + 375*1 = 625*729 + 375 ]Compute 625*729:729 * 600 = 437,400729 * 25 = 18,225Total: 437,400 + 18,225 = 455,625Add 375: 455,625 + 375 = 456,000 ✔️Good.Compute ( V_7 ):[ V_7 = 2V_6 + 3V_5 = 2*456,000 + 3*151,500 = 912,000 + 454,500 = 1,366,500 ]Closed-form:[ V_7 = 625*2187 + 375*(-1)^7 = 625*2187 - 375 ]Compute 625*2187:2187 * 600 = 1,312,2002187 * 25 = 54,675Total: 1,312,200 + 54,675 = 1,366,875Subtract 375: 1,366,875 - 375 = 1,366,500 ✔️Perfect.Compute ( V_8 ):[ V_8 = 2V_7 + 3V_6 = 2*1,366,500 + 3*456,000 = 2,733,000 + 1,368,000 = 4,101,000 ]Closed-form:[ V_8 = 625*6561 + 375*1 = 625*6561 + 375 ]Compute 625*6561:6561 * 600 = 3,936,6006561 * 25 = 164,025Total: 3,936,600 + 164,025 = 4,100,625Add 375: 4,100,625 + 375 = 4,101,000 ✔️Good.Compute ( V_9 ):[ V_9 = 2V_8 + 3V_7 = 2*4,101,000 + 3*1,366,500 = 8,202,000 + 4,099,500 = 12,301,500 ]Closed-form:[ V_9 = 625*19683 + 375*(-1)^9 = 625*19683 - 375 ]Compute 625*19683:19683 * 600 = 11,809,80019683 * 25 = 492,075Total: 11,809,800 + 492,075 = 12,301,875Subtract 375: 12,301,875 - 375 = 12,301,500 ✔️Perfect.Compute ( V_{10} ):[ V_{10} = 2V_9 + 3V_8 = 2*12,301,500 + 3*4,101,000 = 24,603,000 + 12,303,000 = 36,906,000 ]Which matches the closed-form result I got earlier. So, that's a good confirmation.Therefore, the closed-form expression is correct, and ( V_{10} = 36,906,000 ).**Final Answer**1. The closed-form expression for ( V_i ) is ( boxed{625 cdot 3^i + 375 cdot (-1)^i} ).2. The number of votes in the 10th election cycle is ( boxed{36906000} ).